The Concept of Entropy 

in Landscape Evolution 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 500-A 

The Concept of Entropy 

in Landscape Evolution 

By LUNA B. LEOPOLD and WALTER B. LANGBEIN 

THEORETICAL PAPERS IN THE HYDROLOGIC AND 

GEOMORPHIC SCIENCES 

GEOLOGICAL SURVEY PROFESS ION AL PAPER 5 00-A 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962 
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary 

GEOLOGICAL SURVEY 

William T. Pecora, Director 

First printing 1962 

Second printing 1963 

Third printing 1967 

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C~ 20402 - Price 20 cents 
CONTENTS 

Abstract __________________________________________ _ 

General statement _________________________________ _ 

Aspects of entropy ______ . __________________________ _ Open systems _____________________________________ _ 

Principle of least work and entropy __________________ _ 

Longitudinal profile of rivers ________________________ _ 

Demonstration by use of a random-walk modeL _______ _ 

Effect of constraint on stream length _________________ _ 

Pae:e 

A1 

1 

2 

4 

6 

7 

8 

9 

Effect of the constraint of temporal base levels ________ _ 

Probability and the profile of the graded river _________ _ Constraints--summary _____________________________ _ 

Hydraulic geometry------ __ -------------_-----------

The drainage network ______ ---- __ -------_----------- Discussion and geomorphic implications ______________ _ 

]References.----------------------------------------

Page 

A10 

11 

11 

11 

14 

17 

19 

ILLUSTRATIONS 

FIGURE 1. Thermodynamic engine model of steady state ________________________________ _ 

2. Example of a simple statically indeterminate 

truss---------------------------------

3. Sample random walks used in the generation 

of an average stream profile ____________ _ 

4. 1Random-walk profiles subject to constraints 

of stream length ____________ -----------

5. Effect of an absorbent medium on a stream 

profile--------------------------------

TABLE 1. Frequencies of random walks having given 

elevation at various distances from origin, 

in percent, subject to condition that probability of a downward step equals H/6. __ 

2. Frequencies of random walks contained in 

table 1 that reach base level at or before 

distance 10 (figures shown are percent) __ 

Page Page 

FIGURE 6. Portion of random-walk model of stream 

A5 network_----------------------------- A16 

7. Relation of length of streams to stream order_ 17 

6 8. Development of a random-walk drainage 

basin network_________________________ 18 

8 9. 1Relation of number and average lengths of 

streams to stream order ______________ - _ 19 

10 10. 1Re\ation between stream length and drainage area for random-walk modeL _______ - 19 

10 

TABLES 

Page 

TABLE 3. Frequencies of random walks contained in 

table 1 after introduction of an absorbing 

layer (75 percent of random walks absorbed 

Page 

A9 between elev.':l.tion levels 4 and 5; figures 

shown are percent)_____________________ All 

10 

IH 

THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

mE CONCEPT OF ENTROPY IN LANDSCAPE EVOLUTION 

By LUNA B. LEOPOLD and WALTER B. LANGBEIN 

ABSTRACT 

The concept of entropy is expressed in terms of probability of 

various states. Entropy treats of the distribution of energy. 

The principle is introduced that the most probable condition 

exists when energy in a river system is as uniformly distributed 

as may be permitted by physical constraints. From these 

general considerations equations for the longitudinal profiles of 

rivers are derived that are mathematically comparable to those 

observed in the field. The most probable river profiles approach 

the condition in which the downstream rate of production of 

entropy per unit mass is constant. 

Hydraulic equations are insufficient to determine the velocity, 

depths, and slopes of rivers that are themselves authors of their 

own hydraulic geometries. A solution becomes possible by 

introducing the concept that the distribution of energy tends 

toward the most probahle. This solution leads to a theoretical 

definition of the hydraulic geometry of river channels that 

agrees closely with field observations. 

The most probable state for certain physical systems can also 

be illustrated by random-walk models. Average longitudinal 

profiles and drainage networks were so derived and these have 

the properties implied by the theory. The drainage networks 

derived from random walks have some of the principal properties 

demonstrated by the Horto~ analysis; specifically, the logarithms 

of stream length and stream numbers are proportional to stream 

order. 

GENERAL STATEMENT 

In the fluvial portion of the hydrologic cycle a particle 

of water falls as precipitation on an uplifted land mass. 

Its movement to the sea gradually molds a path that 

will be taken by succeeding particles of water. In its 

movement it will do some minute portion of the grand 

task of reducing the land mass in average elevation by 

carrying ultimately to the ocean in solution and as 

transported sediment molecules or particles of the 

continental materials. 

The paths taken by the various droplets that make 

their way, through time, from higher to lower elevations 

represent the drainage network of the land surface, 

the patterns of the channels, and the longitudinal 

profiles of the waterways. As these paths are being 

carved by erosion and solution, the features of the landscape are expressed in the topography of the constantly 

changing surface. 

The paths possible for water and its load have .a large 

variety. There are obviously certain constraints which 

can be identified. Because gravity is the main force 

moving the materials, and works vertically downward, 

each particle of water and its associated load can move 

downward in the fluvial portion of the hydrologic cycle. 

The horizontal distance of movement is governed 

ultimately by the relation between the uplifted land 

mass and the ultimate base level-the ocean. Within 

these two general constraints, however, a large variety 

of paths is still possible. 

Geomorphology is concerned with these paths, with 

the forms assumed during the process of landscape 

evolution, and with the principles governing the development of paths and forms. 

W. M. Davis stated (1909) that the interpretation 

of land forms necessitates consideration of the processes acting, the effects of lithology and geologic 

structure, and the stage in the total evolution of the 

landscape. In attempts to define governing principles, 

geomorphologists have expressed various aspects of 

process and structure in mathematical forn1 by equations of more or less generality. Avl].ilable equations 

have not been found useful, however, as general expressions of spatial and time relations. 

The most general equations which must be satisfied 

are, as in all the physical sciences, the equation of 

continuity which states that matter is not lost, and 

the equation of conservation which states that energy 

can neither be created nor destroyed. 

These laws are so obviously general that they characterize each element, or reach, in any fluvial system. 

Further, the equations must characterize each unit 

of any path and at each instant in time. These two 

equations, then-however necessary-are insufficient 

to explain the paths of particles or the relation between one part of a path to another. Therefore, 

they can alone tell us nothing about the surface form 

of the landscape. Nor can they treat completely of 

the progressive development or change of form with 

time. 

The present paper is an attempt to apply another 

general law of physics to the subject for the purpose 

of obtaining some additional insight into energy distributions and their relation to changes of land forms 

in space and time. 

Al 
A2 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

We acknowledge with warm thanks the helpful 

suggestions and criticisms received from colleagues 

who read an early draft of the manuscript. Particularly we thank Brig. Ralph A. Bagnold, 11. Gordon 

Wolman, John T. Hack, Richard Chorley, N. C. 

Matalas, H. W. Olsen, and Mark Melton. 

Previous work with Herbert E. Skibitzke was very 

influential in the development of ideas included here, 

and we are especially grateful to him for his interest, 

and for the stimulation received from many fruitful 

discussions with him. 

ASPECTS OF ENTROPY 

The thesis of the present paper is that the distribution 

of energy in a river system tends toward the most 

probable state. This principle, somewhat analogous 

to that ilnplied in the second law of thermodynamics 

in relation to thermal energy, governs ultimately the 

paths of movement in the fluvial process and the 

spatial relations between different parts of the system 

at any one time or stage. Further, it is suggested that 

this general principle also tends to govern the sequence 

of development of these paths from one stage in geomorphic history to the succeeding one. 

The development of the landscape involves not 

only the total available energy, but its distribution 

as well, a factor that may appropriately be described 

as entropy, adapting that term from the comparable 

concept in thern1odynamics. 

The first law of thermodynamics is merely a restatement of the principle of conservation of energy when 

heat is included in the energy forms. The second law 

of thermodynan1ics stated in simplest terms is that 

there is an increase in entropy in every natural process 

providing all the system taking part in the process is 

considered. The increase in entropy is a measure of 

the decrease in availability of the energy in the sense 

that a certain amount of energy is no longer available 

for conversion to mechanical work. 

The second law expressed in thermal terms is not obviously related to geomorphic systems in which mechanical rather than heat energy is of principal concern. 

Its applicability will be explained in the discussion that 

follows. 

With the understanding that it is necessary to define 

the system to which the second law of thermodynamics 

is· to be applied, the essential idea to be adapted is that 

the entropy of a system is a function of the distribution 

or availability of energy within the system, and not a 

function of the total energy within the system. Thus, 

entropy has come to concern order and disorder. Information theory utilizes this aspect of the entropy 

concept. 

The order-disorder aspect of entropy may be demonstrated by looking at your desk top. Ours is in disorder. Because of this we need offer no explanation 

of where things are. It is equally probable that a 

given piece of paper is anywhere. If we should take 

the trouble to place things in order, it would be necessary to label the "various piles, and the an1ount of 

labelling (i.e. information) would increase with the 

degree of classification. One can tolerate a certain 

amount of disorder on his desk, but he must continually 

correct this by effort expended in search. This effort is 

putting negative entropy into the desk system; but the 

work done in this effort itself involves a general increase 

in entropy of the environment. 

The degree of order or disorder in a system may be 

described in terms of the probability or improbability 

of the observed state. This aspect might best be stated 

first as a simple example: In a room filled with air the 

individual molecules are moving at random and, because 

of the effects of this movement and the collision of 

molecules, the gas tends to become more or less uniformly distributed throughout the room. Owing to the 

fact that these same collisions occur at random, it is 

physically possible that the random motion of particles 

might create a situation in which all particles of the 

ga.s are for a moment concentrated In one small volume 

in a corner of the romn. The irnprobability that this 

would happen by chance is very great indeed. The 

n1oleculPs of the gas tend to bP.come distributed in a 

random or disorderly way. The chance that given 

degrees of order prevail may be described in the form 

of a probability statement. 

It is quite possible, of course, to pump the gas in the 

romn into a sn1all volu1ne, but work would have to be 

expended on the sy8tem in order to accomplish this 

result.. The overall entropy including that of the external pump would thence be increased. 

The distribution of the energy may be stated in 

tertns of the probability of the given distribution 

occurring relative to alternative distributions possible. 

As expressed by Brillouin (Bell, 1956, p. 159)-

Entropy and probability are practically synonymous for the 

physicist who understands the second principle of thermodynamics as a natural tendency from improbable to more probable 

structures. 

With the statistical conception of entropy in mind, 

the possible application to geomorphic systems becomes 

recognizable. The distribution of energy in a geomorphic system is one way of expressing the relative 

elevation of particles of water .and of sediment which 

gradually will, in the process of landscape evolution, 

move downhill toward base level. The longitudinal 

profile of the river, for example, is a statement of the 
THE CONCEPT OF ENTROPY IN LANDSCAPE EVOLUTION A3 

spatial distribution of streambed materials with regard 

to their elevation and, thus, with regard to their 

potential energy. 

In thermodynamics heat energy is referred to absolute 

temperature as a base. The absolute temperature 

defines an absolute limit or a base datum, the situation 

in which molecular motion becomes zero. It is, then, 

the base level or the datum against which the energy 

content of a thermal system can be measured. 

Systems in geomorphology also have a base datum 

with regard to the distribution of energy. This base 

datum is a datum of elevation, in most cases represented 

by mean sea level. The longitudinal profile of the river 

may be described as the distribution of the potential 

energy of both part.icles of water and particles of 

sediment, as they a.re traversing the fluvial path from 

higher elevation toward base level. The energy distribution n1ay be defined in terms of the probability of the 

occurrence of that particular distribution. This again 

will be developed first for the therinodynamic case. 

If, in a closed thermodynamic system at absolute 

temperature T, E is the thermal energy per unit mass 

of a substance having a specific thermal energy 0 

(energy change per unit of temperature change), 

then a change dE in a unit mass is equal to 

dE=OdT. (1) 

In this situation T may be thought of as being a measure 

of the adverse probability p that the energy exists in 

the given state above absolute zero. Tis not equal to 

p because T is defined in arbitrary units, where p 

is an absolute number constrained between limits of 

zero and 1, but T is a function of p. 

Because thermodynamic entropy is defined as 

then, per unit of mass, 

cp=Of~ 

=O'f~ 

cp=O' log, p+constant 

{2) 

(3) 

where 0' is the specific heat energy in appropriate 

units. 

Thus, entropy in the abstract sense n1ay be defined 

as the logarithm of a probability and may express, for 

example, the ratio of the probability of a given physical 

state to the probability of aU other possible alternative 

states. Such a statistical definition of entropy has 

been used in physical chemistry, information theory, 

and elsewhere (Bell, 1956; Lewis and Randall, 1961). 

An example of possible alternatives would be the 

following: If consecutive single draws are made from 

a deck of playing cards, replacing the drawn card and 

reshuffling after each operation, the probabilities of 

drawing an ace, a face card, or a numbered card would 

differ. The alternative states (aces, face cards, numbered cards) have individual and different probabilities 

of turning up in consecutive draws. The entropy of 

the system is defined not in tern1s of the results of 

any single experiment but in terms of probabilities 

among the alternatives and is of the forin cp=c(log PA 

+log PF+log PN) where PAis the probability of drawing an ace, p, the probability of drawing a face 

card, and PN the probability of drawing a numbered 

card, and c is a constant to convert to appropriate 

units. 

To generalize, then, if a systen1 included various 

alternative states 1, 2, 3, ... n, the individual probabilities of which occurring in various examples are 

PIJ p2, p3 • • • Pn, the entropy of the system is defined 

as the sum of the logarithms of these probabilities 

cp=c'l; log p. (4) 

The probability of a given state, Ph represents the 

fractional chance as compared with unity that any 

example or sample will be in the state designated "1" 

among the n alternatives. 

To explain in another way, if two states, statistically 

independent, of probabilities respectively, Pt and P2, 

and corresponding entropies, cp1 and cp2 from an assemb]y 

of states are con1bined, then the probability of the 

combination is p1 p2 since the probabilities are multiplicative. On the other hand, entropy being an 

extensive property is additive. This relation between 

entropy and probability is satisfied by a logarithmic 

relation as before, thus, 

cp ex: log p+constant. (5) 

In terms of a gas in an isolated system, the Boltzman 

relation between entropy and probability is usually 

stated as 

R cp=N log p+constant (5a) 

where R is the gas constant, N is Avogadro's number, 

and p represents, as before, the probability of being in 

a given state. Because log p is a negative number, 

the sign of cp depends on the difference between R/N log 

p and the constant, representing the base level for 

measuring entropy. 
A4: THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

Since p is a positive number less than 1 equation 

Sa can be written 

or since 

(5b) 

where k is Boltzm.an's constant RJN. In other words, 

the probability of a given energy E, of one state i is 

proportional to the negative exponential of its ratio 

to the total energy kT of all possible states. 

It is now possible to specify the most probable distribution of energy in a system. H the system is 

composed of possible states, 1, 2, 3, •.• n, the individual probabilities of which are Ph PI, Pa, ••• p,.1 

the entropy of the system is maximum when the sum 

~ log p is a maximum. (6) 

This may be written as 

(log Pt+log P2+Iog Pa+ ••• +log p,.)=a maximum. 

(7) 

Because the states 1, 2, 3, ..• n represent various 

alternative states, each represented in the whole 

system, the sum of the probabilities is unity, or 

(8) 

We are now concerned with the values of the individual 

probabilities that would make the sum of their logarithms a maximum. It can be shown that the sum 

~ log p is a maximum when P1=PJ=Pa= ••• p,.. In 

other words, the most probable condition among 

alternatives occurs when the individual probabilities 

of the various alternatives are equal. 

Whether a system can attain this distribution depends on the constraints on the distribution of energy. 

However, in this connection it must be remarked that 

although the quantity Oog·pt+log P2+log .Pa+ ... + 

log Pn] is a maximum when all probabilities are equal, 

this maximum is not a sensitive one. That is, there 

can be some considerable irregularity between the 

several values of the probabilities p, and yet the sum 

of their logarithms may not diverge greatly from the 

maximum value obtained when all probabilities are 

equal. For example, if there are 100 possible states 

and tbP. probability of each state is 0.01, the value 

of l: log, p= -460. If, on the other hand, 50 states 

have a probability of 0.015 and the remaining 50 a 

probability of 0.005, the value of~ log, p=-475, not 

greatly less than that for exactly equal partition of 

probabilities. The geomorphic significance of this 

mathematical fact seems to be that one might expect 

considerable deviations from the theoretic "most 

probable" state even in the absence of marked physica.l 

constraints. Complementary to the flat nature of the 

peak of the curve of the sum of the logarithms, the 

side limbs are very steep, indicating that relatively 

short intervals of time are needed to reach a state 

approaching the maximum probable, although the rate 

of adjustment to the theoretic most probable state 

thereafter may be quite slow if ever achieved. 

OPEN SYSTEMS 

The classical treatment of entropy in thermodynamics deals with closed systems in which entropy continuously increases to a maximum stationary level at 

equilibrium. In closed systems there is no loss or 

addition of energy. Geomorphic processes operate, 

on the other hand, in open systems in which energy 

is being added in some places while in other places 

energy is being degraded to heat and is thus lost insofar 

as further mechanical work is concerned. A river 

system is an example of an open system. Let the 

system be defined as the water and the debris in the 

river channel. As water flows down the channel it 

gives up potential energy which is converted first to 

kinetic energy of the flowing water and, in the process 

of flow, is dissipated into heat along the channel 

margins. 

Precipitation brings increments of energy into the 

system because water enters at some elevation or with 

some potential energy. Heat losses by convection, 

conduction, and radiation represent energy losses from 

the system. Yet, the channel may be considered in 

dynamic equilibrium. 

The steady state possible in an open system differs 

from the stationary state of static equilibrium of closed 

systems. We shall therefore equate the term steady 

state with dynamic equilibrium in geomorphology as 

defined by Hack {1960). For the general case of an 

open system the statement of continuity of entropy is 

as follows (Denbigh, 1951, p. 40): 

rate of increase of entropy in system+ rate of outflow of 

entropy=rate of internal generation of entropy. 

In an open system in dynamic equilibrium, the rate 

of increase of entropy in the system is zero (Prigogine, 

1955, p. 82; Denbigh, 1951, p. 86). The continuity 

equation above then takes the form that the rate of outflow of entropy equals the rate of internal generation of 

entropy. 

In a river system in dynamic equilibrium, therefore, 

between any two points along its length, the rate of outflow of entropy is represented by the dissipation of 

energy as heat. This equals the rate of generation of 
THE CONCEPT OF ENTROPY IN LANDSCAPE EVOLUTION A5 

FIGURE I.-Thermodynamic engine model of steady state. 

entropy which is represented by the energy gradient 

toward base level. At this point it is desirable to have 

an equation for the generation of entropy in a river 

system. For this purpose we shall again consider a 

thermodynamic model depicted on figure 1. This 

model consists of a series of perfect engines, Jh J2, etc., 

operating between heat sources and sinks designated by 

their absolute temperatures, T1, T2, etc. The work 

produced by each engine in unit of time llt is w~, w2, 

etc., which is delivered to a Prony brake which dissipates the work as heat. The working substance is a 

flow of a fluid from temperature source T1 at a rate q1 

carrying a quantity of heat H 1 in a unit of time. The 

amount of work (in heat units) that is done by engine 

J 1 equals, in unit of time !::..t, 

Wt_Ht T1-T2_ (T T) -------ql 1- 2 • ll, ll, T1 

(9) 

It will he noted that the work done by the engine is 

equal to the heat transferred multiplied by the Carnot 

efficiency factor. Further, in thermal units, work done 

in unit time equals flow rate times temperature difference, so the equation is dimensionally consistent. 

Now the heat delivered to sink T2 is 

H2=[Ht-Wd=H1 ~:=q1T2llt. (10) 

At temperature level of T2 let us introduce an additional 

flow of fluid at rate qz carrying a quantity of heat qzT2. 

The total heat therefore carried to engine J2 is H2+H:r 

and the work done in unit time by engine J2 is 

The heat delivered by J 2 to T3 is 

Ha=H2+H:~:-~=(H2+Hz) ~:· 

Hence, the heat delivered to each sink decreases in 

proportion to the ratio of the absolute temperatures of 

sink and source. 

A unit of work done by engine J 1 and dissipated by 

the Prony brake, when divided by the absolute temperature, represents a generation of entropy or a unit 

export of entropy from the system. This work per 

unit of time is:~, accomplished at temperature T11 the 

quotient of which represents the change of entropy per 

unit of time or 

WI. _!.=(l::..r!>) At T At 1 
A6 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

and from equation 9 

and 

Similarly at engine J 2 

(11) 

Thus, in each engine the rate of production of entropy 

is proportional to the rate of flow of fluid, the temperature difference between source and sink, and inversely 

proportional to the absolute temperature. 

Hence, in more general terms, if in time dt the fluid 

flows a distance dx, which represents the distance apart 

of the adjacent engines, operating between a corresponding temperature difference dT 

d¢. dT 1 

dt=qdxT 

or in terms of entropy per unit of flow 

d¢/dt dT 1 

-q-=dxT 

(12) 

(13) 

Thus the right-hand side of equation 13 may be 

interpreted as describing the most probable distribution 

of energy in the thermodynamic model. This distribution is such that energy is in inverse proportion to 

temperature above base level T=O. Equation 13 is a 

thermal statement having no direct relationship to a 

river system. There are, nevertheless, several analogous 

points, which, if supported by field evidence, can lead 

to further inferences about river systems. 

The analogy to the river system is that each engine 

represents a reach, !l.x, of the river which receives at 

its uppeF end a flow of water at total head Ht, and 

delivers at its lower end the same flow at H2• The 

difference in head is degenerated into heat which, like 

the Prony brake, exerts no influence on the transport 

of energy or mass in the river. Within the river reach, 

as within the engine, certain reversible reactions may 

take place, as for example transfers between kinetic 

and potential forms of energy as the flow expands or 

contracts. These, too, may be necessu.::."y adjuncts to 

the operation of the engine or the transport of water 

in a river, but being isentropic, have no effect on the 

equation of continuity of entropy. 

If one interprets equation 13 as stating that the most 

probable distribution of energy in the engine model is 

in inverse proportion to temperature above base level 

T=O, then a corresponding statement for a river system 

is 

d¢' fdt dHfdx 

---cr-=ll {14) 

where cf/ is the entropy in the special sense of the river 

system, H is elevation or total energy content above 

base level, and Q is rate of river discharge. Again note 

that tributary entrance does not alter the relationship 

because equation 14 is written in terms of rate of 

entropy change per unit of flow rate. 

According to Prigogine (1955, p. 84), in the evolution 

of the stationary state of an open system the rate of 

production of entropy per unit volume corresponds to 

a minimum compatible with the conditions imposed on 

the system. In the case of the engine model, this means 

that if a stationary state prevails, the work done by each 

engine (the source of the entropy production) is a minimum. Hence, a stable system corresponds to one of 

least work, a point to which we refer in the next section, 

after which we shall examine the distribution of the 

production of entropy in a river system. 

PRINCIPLE OF LEAST WORK AND ENTROPY 

There is therefore considerable logic to the adage 

that nature follows the prir1Ciple of least work. The 

idea is well developed in the analysis of stresses of a class 

of framed structures where the principles of statics are 

insufficient to determine the division of stresses. These 

structures are, therefore, called "statically indeterminate," and illustrate how inferentially the principle 

of maximum entropy is used to complete the solution. 

Consider the simple truss on figure 2. The crossed 

w 

FIGUBE 2.-Example of a simple statically indeterminate truss. 

braces make this a common form of statically indeterJninate truss. The principles of statics give 10 equations, but there are 11 unknowns. The civil engineer 

provides the necessary additional equa.tion by introducing the assumption, first applied by Castigliano in 

1879, that the stresses are distributed so that the total 

strain energy in the several members is a minimum. 

The strain energies are defined as F'2 Lj2EA where F 

is the total stress on each member, Lis its length and A 

its cross-sectional area, and E is the modulus of elasticity of the material. This is called the principle of 

least work. 
THE CONCEPT OF ENTROPY ~ LANDSCAPE EVOLUTION A7 

In structural engineering no attention has been paid 

to the fact that the distribution of stresses so determined is also the most probable. 

We shall demonstrate the equivalence of the principle 

of least work with that of maximum probability by 

application of the principles just discussed. As in the 

river profile, an articulated system such as a truss has 

an entropy equal to 

cp=~ log p+constant. 

The lllOSt probable distribution emts when cf> is a 

rna:x'i.mum. Let the ratios En/E, E~/E, En3/E ... , 

etc., represent the proportional division of the strain 

energy among the several mmnbers, where E is the total 

strain energy in the system-the product of the weight 

lV times deflection. (Note the analogy to the discussion leading to equation 5 and immediately following 

equation 5b.) 

However, the joint probability that a particular 

combination of strain energies exists among the several 

members is 

Where there are no physical limitations, this joint 

probability would be a maxi1num when all probabilities 

are equal; but in this case, conditions of statics set 

constraints on the values of the strain energi£'s so that 

we can state only that the most probable combination 

exists when En +En +En~+ ... is a. minimum. 

This is exactly equivalent to the statement of least 

work; but the principle of }past work must be recognized 

as only one species of a larger class or genus including 

all of the states of n1aximum probability. 

In the structural exan1ple the laws of statics left 

only one degree of freed01n to be met by the principle 

of maximum probability. No general solution could 

be made ba..-,ed entirely on one principle--that is, entirely on the basis of statics, or entirely on the basis of 

maximum probability. Similarly in geomorphic problems, the set of physical faet.ors includes rnany variables 

such as the amount of water and sediment to be carried, 

the fluid friction, and the river transport capacity. 

The equatlons connecting these factors leave several 

degrees of freedom remaining. In other words, a river 

system is "hydraulically indeterrninate." A river can 

adjust its depth, width, or velocity to a given slope in 

several ways so that it is necessary to establish the 

river profile and the hydraulic geometry on the basis 

of maximum probability. 

The general implication for rivers may then be 

stated as follows: The principle of least work is one of 

several ways in which the condition of maximum 

probability may be satisfied. The river channel has 

the possibility of internal adjustment among hydraulic 

variables to meet the requirement for maximum 

probability, and these adjustments tend also to achieve 

minimiza.tion of work. In systems other than rivers 

wherein the adjustment is actually to a condition of 

least work, maximum probability is achieved. The 

geomorphologist's intuitive inference that river equilibrium is a condition of least work (see, for example, 

Rubey, 1952, p. 135) 1s not complete. As will be 

~hown in the section on hydraulic geometry, other 

factors restraiR the system from attaining a state of 

least work. 

LONGITUDINAL PROFILE OF RIVERS 

A concept developed in a preceding section states 

that in an open system the distribution of energy tends 

toward the most probable. In the open system represented by the river, the unit under consideration is a 

unit of length along the river channel. The concept 

applied to the river system yields the result that the 

most probable distribution of energy exists when the 

rate of gain of entropy in each interval of length along 

the river is equal. 

To develop the reasoning leading to that conclusion, 

refer back to equation 14, 

dq,' 1 dH 1 

dt Q= dx H (l4) 

The rate of production of entropy in a river system is 

a;;, and per unit of mass volume rate (discharge) is 

1t' · ~· This quotient is inversely proportional to the 

energy content above base level, H, and directly proportional 4-o the loss of potential per unit of distance, 

dH 

dx· 

The right side of equation 14 is a statement of the 

proportional distribution of energy relative to base 

level, H =H0• Then it also can be considered a statement of the probability of a given distribution of 

energy between H =H0 and the maximum value of H 

in the system in question. 

The probability of a particular combination of 

values of energy content of unit distances along the 

course of the open system of the rivPr would be similar 

in form to equation 15, or 

where dHJ dH2 h 1 f h d . dx , dx , etc., represent t e oss o ea m 

successive units of length along the river length. 
AS THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPmC SCIENCES 

This joint probability would be maximun1 when all 

the probabilities are equal. Because in each unit 

length, dx, the loss of head, dH, would have a given 

probability, p 

dH1 1 

dx H ocp1 

etc., and the joint probability is greatest when p1 =p2=p8 

h dH1 dH2 dHs dH . . p., t en dxH=dxH= dxH or dxH IS equal tn 

each unit of length along the river and thus is constant. 

Also, then, the time rate of change of entropy per unit 

of flow rate is constant. Because 

dH 1 

dx H=constant, (16) 

integrating 

(17) 

where a and b are coefficients and 0 represents the base 

level. 

To sum1narize, the most probable sequence of energy 

losses in successive units of river length corresponds to 

a uniform increase of entropy in each unit length. 

When this specification is fulfilled without constraint 

on length, the longitudinal pr'lfile tends to become 

exponential in form, a re.snlt. in n.greement WJth manv 

actual river profiles. · 

How('V~r, it, may not be clear how the rate of loss 

of energy at any point is necessarily related to e.lPvation 

above ba.se level. We refer back to our statement that 

entropy must represent a ratio between a givrn state 

to the number of alternate states. Note that dh/d:j 

is the gradient of energy or river slope, S, and then 

(18) 

The quantity S= a:: is the rate of energy loss and 

represents the given state. Then the quantity fl n1ust 

represent the range of choice of the rate of loss of 

energy at any level. The probability p then represents 

the proportion of the total elevation lost as the river 

approaches ba.c;e level. This we assert to be implicit 

in the Rta.tement that the higher the landscape above 

base level, the greater becomes the distribution of 

possible slopes. This range must decrease as one 

approaches base level; in the same way the possibility 

of orderliness an1ong n1olecules of a gas becomes less 

and less as temperature is lowered and becomes zero 

at the base level of zero absolute temperature. 

As demonstrated with the engine model example, 

the most probable state is one in which the rate of 

energy expenditure per unit mass of fluid is proportional to the height above base level. Because the 

statement applies per unit mass of fluid, it will apply 

throughout a river composed of a branched network 

of channels of increasing size downstream which collect 

and convey water and sediment. It is only necessary 

to postulate that water and sedi1nent once entrained 

remain in transit; i.e., there are no losses in transit 

and there is no constraint on length. 

DEMONSTRATIONS BY USE OF A 

RANDOM-WALK MODEL 

The equations state that a profile of energy distribution in the open systmn represented by a river 

channel should be exponential in form and that this 

profile corresponds to the most probable one under 

the conditions stated in the previous section-that is, 

no constraint on length. This is demonstrated by a 

random-walk model constructed in the following way: 

Consider a point on the surface of a landscape above 

base level as shown in figure 3. The flow moves in 

5 

4 

z 3 0 

t= 

~ 2 LLI 

..J 

LLI 

/ Initial point 

x } Sample random 

0 walks 

-x-x-x 

\x-~/ Average profile 

~~"LL /\ H=5C5/6>x 

\ -x-x~x-x~~x-x-x 

5 10 

DISTANCE, X 

\ 

15 

FJGURE 3.-Sample random walks used in the generation of an 

average Rtream profile. 

unit steps to the right with two choices at each step. 

There is the probability p that it will move downward 

one unit of elevation or the probability q that it will 

continue at the sa1ne level. Since there are only these 

two alternatives, p+q must equal unity. The condition that the rate of energy expenditure remains proportional to height above base level is included in the 

model by setting the probability p of a downward step 

proportional to the height above base level, decreasing 

to zero when a random walk reaches base level. 

In this model, figure 3, an initial point was set at an 

elevation of 5 units above base level. Since there are 

6levels (0, 1, 2, 3, 4, and 5), the probability of a downward step is H/6. Thus, the probability of a downward 

step decreases with elevation. Hence, random walks 
THE CONCEPT OF ENTROPY ~ LANDSCAPE EVOLUTION A9 

were constructed as follows: A pack consisting of 

5 white cards and 1 black card was prepared. The 

number of cards represents the total number of levels, 

and the number of white cards in the pack represents 

the elevation above base level, in this case initially 5. 

The cards were shuffled and a card selected at random 

and then replaced by a black card. If a black card is 

selected and replaced by a black, there is no change 

in elevation. If a white card is replaced by a black 

card, then elevation decreases by one unit. This 

process is continued and a record made of the number 

of white cards in the pack at each step. The probability of choosing a white card (equivalent to a downward step) decreases as the proportion of white cards 

in the pack decreases. When all white cards have 

been replaced by black cards, the random walk has 

reached base level. 

Table 1 lists the frequency of random walks at 

different elevations and distances constructed by 

this model, and figure 3 shows the results of two of 

these random walks. The graph of the equation 

defining the mean position of all possible walks generated by the above model is also plotted on figure 3. 

The equation of this line is 

(19) 

Referring back, equation 18 states that 

dH 1 

dx H ex: log p. 

Integrating 

(20) 

H0 is height of an initial point above base level. The 

equation is that of a longitudinal profile expressed in 

terms of a statistical parameter. 

Equation 20 is then a more general form of equation 

19. The probability p in equation 20 corresponds to 

the fraction 1!5

, or more generally, H!1+H. In 

each unit of distance, x, the elevation declines by the 

ratio H/1 +H. In other words, p in equation 20 equals 

the probability that the rate of loss of energy in a 

unit length of river is a fixed proportion of the total 

head above base level. This probability is a constant. 

The profiles of many major streams do indeed follow 

an exponential law as is well known (Schoklitsch, 1933). 

There is no point in attempting to explore or compare 

actual profiles with the exponential law because, as we 

shall show, the most probable profile under various 

constraints differs from the simple exponential one. 

Analyzing the possible agreement of specific examples 

TABLE I.-Frequencies of random walks having given elevation at 

flarious distances from origin, in percent, subject to condition that 

probability-of a downward step equals H/6 

Elevation 

(H) 

Distance 

o 1 2 3 4 5 6 1 8 9 ro 

------1·--1-------------------- 5 _____________________ 100 17 3 ----- ----- ----- ----- ----- ----- ----- ----- 4 _____________________ ----- 83 42 17 6 2 ----- ----- ----- ----- ----- 3 _____________________ ----- ----- 55 56 39 24 14 7 3 2 1 

2--------------------- ----- ----- ----- 27 46 50 45 37 29 20 14 1 _____________________ ----- ----- ----- ----- 9 23 36 45 50 52 50 

o _____________________ ----- ----- ----- ----- ----- 1 s 11 18 26 35 

Averageelevation •••• 5.0 4.17 3.48 2.00 2.42 2.02 1.68 1.40 1.17 0.98 0.81 

with the theory becomes a problem of identifying the 

several influences upon the actual profiles-which is 

not the objective of this theoretical paper. 

EFFECT OF CONSTRAINT ON STREAM LENGTH 

The exponential form of stream p.LJfile is the most 

probable when there are no constraints on channel 

length; the profile approaches the base level asymptotically. If, as in mountainous headwater streams, the 

constraint on stream length greatly exceeds the constraint exerted by base level, then the probability of a 

vertical step decreases downstreamward with respect 

to distance rather than with respect to elevation as 

before. The resulting profile is therefore the converse 

of the preceding case. In other words, height and 

length are exchanged in their relationship; thus 

x=L(p)8 (21' 

where p=L/1+L, L is total stream length, and x is 

distance downstream from the headwaters. Equation 

21 also defines an exponential profile, but with respect 

to distance rather than with respect to elevation as in 

the previous case. In other words, the elevations vary 

with the logarithms of distance. In this case, His the 

height above the elevation where x equals L. The 

profile is not asymptotic to this level. 

Equation 21 can be written in the form Hoc log, x/L. 

Profiles of the form defined by this equation were found 

by Hack (1957, p. 70 and following) in many headwater 

streams in Virginia and Maryland. 

The effect of a constraint on length upon the developInent of a profile may be illustrated by a random-walk 

model. In this case, one limits the randon1 walks to 

those paths that. reach base level at or before reaching 

a given limiting distance. For example, according to 

the frequencies listed in table 1, 35 percent of all random 

walks developed in the model used to define figure 3 

reached base level at or before distance 10. The 

frequency distribution of these walks is listed in table 2, 

and the average elevation as calculated from these 

frequencies is plotted on figure 4. For comparison there 

is also plotted on figure 4, the profile for the random 
AlO THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

5 

l: 4 

z 

0 3 

i= 

< 

> 2 liJ 

..J 

liJ 

0 

0 5 

DISTANCE. x 

10 

FIGURE 4.-Random-walk profiles subject to constraints of 

stream length. 

walks that reached base level at or before distance G, 

and the profile for all random walks (that is, without 

restriction as to length) as shown in figure 3. The 

concave profile is evident on all, but none of the 

restricted profiles is tangent to base level and the angie 

at which the profile intersects base level increases with 

increasing constraint on stream length. 

This result leads us to judge that constraints of 

length tend to diminish the influence of base level on 

the profile, and so to impair the proportional distribution of head loss in a river system. Other factors a.re 

operative, of course, and their influence must also be 

investigated. Some of these other factors are hydraulic. Their interaction is also illuminated through 

application of the probability principle, as will be shown 

in a subsequent section where a theoretical derivation 

of the hydraulic geometry of rivers is presented. 

TABLE 2.-Frequencies of random walks contained in table 1 that 

reach base level at or before distance 10 (figures shou:n are percent) 

Elevation 

Distance 

o 2 3 4 5 6 7 8 9 ro ------1·--1--1----------

5--------------------- 35 4. 7 0. 6 ----- ----- ----- ----- ----- ----- ----- ----- 4 _____________________ ----- 30.3 12 3. 5 0. 7 ----- ----- ----- ----- ----- ----- 3 _____________________ ----- ----- 22.4 18 9. 3 3. 8 1. 2 0. 2 ----- ----- -----

2--------------------- ----- ----- ----- 13.5 19 16.4 10.3 5 1. 6 ----- ----- 1 _____________________ ----- ----- ----- ----- 6. 0 13.8 18.5 18.8 15.4 8. 7 -----

:~:-~~::~;,:;:::: 7o ~~~=;=;;+..+,.%.so~ *I~ 

EFFECT OF CONSTRAINT OF TEMPORAL BASE LEVELS 

Only the sea is a perfect sink-an absolute base levelin the geomorphic sense. There are other base levels, 

but all these must, in the scope of time and place, he 

considered temporary or imperfect. Base levels may 

be as temporary as a bar of gravel deposited in a channel by a flood from a tributary; base levels may be as 

imperfect as a riffie which transmits down11tream only 

part of the energy of the flowing water and absorbs the 

remainder, and which are drowned out at high river 

stages. The valley floor is an example of an imperfect 

base level relative to streams draining to it from the 

valley sides. This is especially true in an arid region 

where both water and sediment are lost in the valley 

alluvium in small floods, but carry through the valley 

in large floods. 

One can set up a random-walk model of imperfect or 

temporal base levels, by interposing an imperfect absorbing medium in the random walks. As Feller (1950, 

p. 279) explains, random walks may be subject to absorbing barriers or reflecting barriers. All random 

walks terminate at an absorbing barrier. The latter 

have a geomorphic significance. In the random-walk 

model of the river profile, the sea becomes a perfect 

absorbing barrier. However, the effects of a temporal 

base level as described may be modeled by a partial 

absorbing barrier, which "absorbs" a given percentage 

of the random walks and permits the remainder to pass 

through unaffected. 

5 

l: 4 

z 

0 3 

i= 

< 

> 

LLI 2 ..J 

liJ 

0 

0 5 10 

DISTANCE. X 

FIGURE 5.-Effect of an absorbent medium in a stream profile. 

Figure 5 shows the results of such a calculation using 

the same random-walk model as used in the development of figure 3. The absorbing medium was placed 

between elevations 4 and 5 and it was assumed to 

absorb 7 5 percent of the random walks; that is, the 

frequencies at elevation 4 and below, as listed in table 1, 

were reduced by 75 percent. The data from which 

figure 5 was derived are presented in table 3. Average 

elevations at the various distances were computed 

accordingly. The resulting profile plotted on Cartesian coordinates, figure 5, is convex in its upper 

part leading in to an exponential cn.rve. This profile 

appears comparablP. to a typical hillside profile or 

profile of certain headwater draws or rills. This 

suggests that a tributary strearn to which a hill slope 

or draw is graded is a partial or imperfect base level 

relative to the hill slope or draw. 

The absorption factor does not mean that 7 5 percent 

of the water and sediment is absorbed-we refer, 

rather, to the transmission of the influence of lower 
THE CONCEPT OF ENTROPY IN LANDSCAPE EVOLUTION All 

base levels. The complement of the absorption factor 

describes the transmission of the influence of base level 

upon higher elevations. 

TABLE a.-Frequencies of random walks contained in table 1 

after introduction of an absorbing layer (75 percent of random 

walks absorbed between elevation levels 4 and 5; figures 

listed are percent) 

Distance 

Elevation 0 2 a 4 5 6 1 8 9 ro 

---·---1---1-- --------------1-~ 

5--------------------- 100 17 3 ----- ----- ----- ----- ----- ----- ----- ----- 4 _____________________ ----- 21 10 4 1 a ______________________________ 14 14 10 _6 ____ 3 ____ 2 ___ ----- ----- ----- 2--------------------- ----- ----- ----- 7 11 12 11 9 _7 ____ 5 ____ 3 __ _ 1 _____________________ ----- ----- ----- ----- 2 6 9 11 12 13 13 o_____________________ _____ _____ _____ _____ _____ _____ 1 3 4 7 9 

Average elevation____ 5. 0 4.45 3. 60 2. 90 2. 42 2. 02 1. 68 1.40 1.17 0. 98 0. 81 

PROBABILITY AND THE PROFILE OF THE GRADED 

RIVER 

The probability nature of the profile also carries certain implications about deviations from the mean or 

most probable profile. Departures from the exponential 

form represent increasing degrees of improbability. 

Variations in the profile due to changes in rock type or 

even of stream length will persist for a short or long 

period of time, depending on the scale of the constraint. 

However, local departures from a general rule of a proportional decrease in energy expenditure, when considered in the time scale of physiographic processes, are 

ephemeral elements. This may be true even when the 

irregularities are as conspicuous as lakes, waterfalls, or 

torrents. This conclusion merely affirms a view long 

held in geomorphology. 

A contribution made by the entropy concept is that 

the "equilibrium profile" of the graded river is the profile of maximum probability and the one in which 

entropy is equally distributed, constituting a kind of 

isentropic curve. 

CONSTRAINTS-SUMMARY 

It may be visualized that the study of the landscape 

is the study of constraints imposed by geologic structur~, lithology, and history. To identify those constraints is the purpose of much of geologic inquiry; to 

eva]uate the effects of constraints upon the landscape is 

an object of quantitative geon1orphology. The way in 

which some constraints affect the river profile can be 

evaluated if one considers the profile to approximate its 

maximum probable condition under a given set of constraints. Those treated in this paper are summarized 

below. 

1. The absence of all constraints leads to no solutionan obvious but not a trivial conclusion. 

2. The longitudinal profile of a stream systern subject 

only to the constraint of base level is exponential with 

respect to elevations above base level. 

3. The profile of a stream subject only to the constraint 

of length is exponential with respect to stream length 

(a logarithmic function with respect to elevation). 

4. The constraint of length added to that of base level 

decreases the concavity and diminishes the influence 

of base level. 

5. Introduction of the constraint of a partial base level 

above that of the sea adds a measure of convexity in 

the profile. 

HYDRAULIC GEOMETRY 

The use of a probability statement to cornplete the 

solution of a set of hydraulic equations can be illustrated thro.ugh evaluation of the hydraulic geometry of 

river systems. In attempting to explain the morphology of rivers that are authors of their own geometries, 

to introduce additional hydraulic relationships adds 

more unknowns. Usually their nun1ber exceeds the 

number of equations and a direct solution becomes 

impossible. For example, Leopold and Maddock (1953, 

p. 50) outline a set of eight conditions that n1ust be 

met to evaluate the hydraulic geometry of a river 

system. There are not as many as eight known interrelationships between these variables. In the following 

simplified example, evaluation of only five unknowns 

(velocity, depth, width, slope and friction) is sought. 

However, only three equations connecting these variables :1re available. The remaining conditions will be 

satisfied by a statement of maximum probable distribution of energy in the system-the basic concept 

implied in the minin1um production of entropy in an 

open system. 

Leopold and Maddock (1953) describe and evaluate 

from field data the hydraulic geometry of river channels 

by a set of relations as follows: 

vocQm 

DocQ1 

wocQ11 

scx:Q• 

nocQ' 

where v is the mean velocity, Dis the mean depth, w 

is the surface width, and sis the energy slope, and n is 

the friction factor at a cross section along a river 

channel where the rnean discharge is Q. It is desired 

to evaluate the exponents in a downstream direction 

as discharge of uniform frequency increases. 

Smne of the principles we have described can provide 

estimates of the magnitude of the exponents of the 

above relationships. The exponents m, j, b, z, and y 

describe the variability in velocity, depth, width, slope, 

and friction along a river channel, but do not uniquely 

determine the ntagnitudcs of these properties. 
A12 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

The first condition is that specified by the equation 

of continuity Q=vDw, which requires that 

m+J+b=1.0. 

A second condition is introduced by the Manning 

equation for hydraulic friction. This requires that 

velocity, depth, energy slope, and friction factor have 

the following relationship: 

vocDisljn. 

Since 

Hence 

The third condition introduced is that the transport 

of sediment per unit mass of water is uniform. This 

is also a reasonable assumption and is a close approximation to the conclusion reached by Leopold and 

Maddock from analysis of suspended sediment transport data (1953, p. 26). Although many formulas for 

sediment transport have been devised, most can be 

expressed in terms of stream power as suggested by 

Bagnold (1960). Power is an important factor in the 

formulation of the hydraulic geometry of river channels. 

As explained by Bagnold, the stream power at :Bows 

sufficiently great to be effective in shaping the river 

channel is directly related to the transport of sediment, 

whose movement is responsible for the channel 

morphology. 

Laursen (1958, p. 198), commenting on the general 

unsatisfactory nature of our knowledge of transport 

phenomena, lists several typical equations for the 

transport of sediment, based on flume experiments, 

that have been reported in the literature. The average 

relation shows sediment transport in excess of the 

point of incipient motion to vary about as (vDs) 1

• 5 

where vDs is the stream power per unit area. In terms 

of sediment per unit discharge, that is the concentration, 0, the several equations average out as Oocn 

(vD) 0

·

5

s1

•

5

, a result that is consistent with the conclusion reached by Bagnold (1960). There is in addition to be considered the effect of sediment size. 

Examination of several equations indicates that sediInent transport varies inversely as about the 0.8 

power of the particle size. There have been several 

attempts to relate particle size to the friction factor n; 

at this point we shall use the Strickler relation thaL 

the value of n varies as the ~ power of the particle 

size. It is realized full well that both the sediment 

transport and the friction factor are influenced by 

many other factors such as bed form and the cohesiveness, sorting, and texture of the materic:tJ. These are 

the kinds of influences, themselves effects of the river, 

that prevent a straightforward solution of river 

morphology. In order to limit the number of variables 

only the effect of particle size on transport will be 

considered, as this factor varies systematically along 

a river from headwater to mouth. Thus, sediment 

transport used in this study reads Ooc (vD) 0

·

5

s1

·

5

/n4

• 

The sediment transport per unit discharge in the 

river system will he recognized as a hydrologic factor 

that is independent of the hydraulic geometry of a 

river in dynamic equilibrium. Consequently sediment 

concentration may be considered constant, and therefore 0.5m+0.5f+ 1.5z-4y=O. 

Thus, there are three equations: continuity, hydraulic friction, and sediment transport. There are 

five unknowns. The two remaining equations will 

be derived from a consideration of the most probable 

distribution of energy and total energy in the river 

system. 

The probability of a given distribution of energy 

is the product of the exponential functions of the 

ratio of the given units to the total, as in equation 15. 

The ratios of the units of energy Eh E2, etc., representing the energy in successive reaches along the 

river sufficiently long to be statistically independent, 

. Et E2 

to the total energy E m the whole length, are E' E' 

. . . ~- The product of the exponentials of 

these is the probability of the particular distribution of 

energy or, as was seen in equation 15, 

where n remains finite. 

As previously, the 1nost probable condition is when 

this joint probability, p, is a maximum and this exists 

when E1=E2=E3 ••• =En. Thus energy tends to be 

equal in each unit length of channel. 

Equable distribution of energy corresponds to a 

tendency toward uniformity of the hydraulic properties 

along a river system. Consider the internal energy 

distribution. Uniform distribution of internal energy 

per unit mass is reached as the velocity and depth 

tend toward uniformity in the river system. Uniformity of velocity and depth requires that the exponents m andj be as near zero as possible, and therefore according to the Manning equation ~z-y should 

approach zero. 

Since the energy is largely expended at the bed, 

equable distribution of energy also requires that 

stream power per unit of bed area tend toward uniformity, that is, that Qsjw tend toward uniformity. 

This condition is met as the sum of m+f+z tends 
THE CONCEPT OF ENTROPY IN LANDSCAPE EVOLUTION A13 

toward zero. An opposite condition is indicated by 

Prigogine's rule of minimization of entropy production 

that characterizes a stable open syE&tem and which, 

therefore, leads to the tendency that the total rate of 

work, l;QsaQ in the system as a whole be a minimum. 

Because sex(!, then ~QI+'AQ-7a minimum. For a 

given drainage basin this condition is met as z takes 

on increasingly large negative values. However, 

there is a physical limit on the value of z, because for 

any drainage basin the average slope ~sAQJl;AQ must 

remain finite. This condition is met only for values 

of z greater than -1, and therefore z must approach 

-1 or 1+z approaches zero. 

To summarize, we have introduced three statements 

on the energy distribution1 2Z-y-70 

m+f+z-70 

(1+z)-70. 

The condition of minimum total work tends to make 

the profile concave; whereas the condition of uniform 

distribution of internal energy tends to straighten the 

profile. Hence, we seek the most probable state. 

The most probable combination is the one in which 

the product of the probabilities of deviations from 

expected values is a maximum. It is unnecessarv t.o 

evaluate the probability function, provided one can 

assume normality, as we can then state directly that 

the product of the separate probabilities is a maxiTnum 

when their variances are equal. 

where F11 F2, Fa represent the several functions stated 

above. Therefore, 

for which there are two possible solutions: 

or 

of which only one leads to a possible result. 

The absolute values of the standard deviations need 

not be known, as we can infer their relative vahle~, 

For example, letting F1 = ~z-y, the standard deviation 

of F1 is 

and 

F2=m+J+z; u,2= ..Ju,.2+ui+u,2 

F3=(1+z); u,3=u,. 

The standard deviations ,,., u" u,, and u, represent the variability of the several factors as may 

occur along a river system. In this solution, they are 

taken to be proportional to their respective values. 

Since these values are not known initially, the problem 

must be solved by iteration. Fortunately, the solution 

is not sensitive to the values of the several standard 

deviations, so the solution converges rapidly. No 

knowledge of their values is available for the first 

trial solution so all values are considered equal. Thus: 

The results of the first solution are then used to derive 

first estimates of the standard deviations, and a second 

computation carried forward. A third calculation is 

made to confirm the results of the second trial solution. 

These three equalit.ies lead to two independent equationsy=-~ (m+f) 

z=-0.53+0.93y 

which, together with the three hydra.ulic conditions 

m+J+b=l.O 

m=%m+~z-y 

~m=~J+1.5z-4y=O, 

lead to a solution for m,j, b, z, andy. The final values, 

derived without reference to field data, are compared 

below with others obtained previously from analysis of 

field data on actual rivers. 

Values of exponents of the hydraulic geometry, in downstream directi011 

Average values from data on rivers in midwestern 

Theoretical values, this paper United States 

Velocity, m= 0. 09 

Depth, f = . 36 

Width, b = . 55 

Slope, z =-. 74 

Friction, y = -. 22 

1 0. 10 

•. 40 

•. 50 

1-.49 

1 Leopold and Maddock, 1953, p. 16. 

1 Leopold, 1953, p. 619, from practically same river data used by Leopold and 

Maddock and therefore considered comparable. 
AI4 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

Because velocity is such a large factor in energy expenditure, the value of m, the exponent of velocity, is 

close to zero, making this characteristic the most conservative. 

The value of z (the measure of the variability in 

slope) is least satisfactorily defined from field data, as it 

is affected by variation in the friction factor. Thus, the 

value of z as reported above is -0.53+0.93y. Hence 

the value of z is -0.53 where grain size is constant. 

The value of z was found by Leopold (1953, p. 619) 

to be -0.49 for midwestern rivers. Henderson (1961) 

shows that for a stable channel of uniform grain size 

the slope varies as the -0.46 power of the discharge. 

However, when a large variety of river data were averaged, including data on ephemeral channels, Leopold 

and Miller (1956, p. 25, 26) obtained a value of z averaging near -0.95. This latter average is probably 

less comparable with the Leopold-Maddock data than 

the average value Leopold (1953) compiled frorn nearly 

the same list of midwestern rivers. Values of z in excess 

of 0.50 may be attributed to the common tendency for 

the friction factor to decrease downstream (increasing 

discharge) as particle size decreases. In canals, for 

example, where y is positive, the value of z is less than 

0.50. 

This study of hydraulic geornetry develops a method 

of solution through introduction of probability statements that we consider to be sound in principle. Though 

the results obtained by this theoretical derivation of exponents in the hydraulic geometry do agree quite well 

with field data, we are far from satisfied with some of 

the physical relationships. For example, there is uncertainty whether the transport equations represent 

physical relationships or are in fact formulas for sediment transport under regime conditions. Several alternative assumptions were tried and the resulting exponents also agreed quite well with field data. But the 

present state of knowledge about some of thf\se interactions among the hydraulic and sediment factors allows 

various reasonable but empiric assumptions that, nevertheless, yield results of the same order of magnitude. 

THE DRAINAGE NETWORK 

As has been shown in the example of the longitudinal 

profile, the probability that a random walk will fall in 

certain positions within the given constraints can be 

ascertained. There is also a mean or most probnblc 

position for a random walk within those constraints. 

This statement suggests the possibility that a particular 

set of constraints might be specified that would de·· 

scribe the physical situation in which drainage channels 

would develop and meet, eventuating in the drainage 

network, 

Let us postulate that precipitation falling on a uniformly sloping plain develops an incipient set of rills 

near the watershed divide and that they are oriented 

generally downhill. As the rills deepen with time, 

crossgrading begins owing to overflow of the shallow 

incipient rills in the manner postulated by Horton (1945, 

p. 337). The direction that the crossgrading takes place 

and the micropiracy of incipient rills is, as Horton implies, a matter of chance until the rills deepen sufficiently to become n1aster rills. 

This randomness in the first stages of crossgrading 

1night be approximated in the following conceptual 

model which is amenable to mathematical description. 

Consider a series of initial points on a line and equidistant frmn one another at spacing a. Assume randon1 

walks originating at each of these points. In each unit 

of time each random walk proceeds away from the 

initial line a unit distance. But let us specify that each 

walk may move forward, left or right at any angle, but 

may not 1nove backward. The accmnulation of moves 

will produce in time sufficient cumulative departures 

from the orthogonal to the direction of the original line 

of points that some pairs of paths might meet. After 

such a junction only one walk proceeds forward in like 

manner and would behave similarly, just as when two 

stream tribut.a.ries join the single stream proceeds 

onward. 

The physical situation described is analogous to the 

statistical model called the "gambler's ruin" (Feller, 

1950, chap. 14). This model treats the }Jrobability 

that in a certain nmnber of consecutive plays, a gambler 

playing against the "house" will lose all of his ca.pital to 

the house. In the model, the capital assets of the 

gambler and the house are specified. If the ga1nbler has 

a lucky run he accumulates capital at the expense of the 

house. If the reverse takes place, he gradually may 

lose his capital completely. 

The statistical statement of the probable duration of 

the game (number of plays) is given by D=Z(A-Z) 

provided the probabilities of a win and loss are equal, 

and where Z is the capital of the gambler, and A is the 

total capital in the game (house plus gambler). The 

size of Z and A are measured in units of the size of 

equal wagers. 

The analogy to the physical situation of the postulated developing drainage network may now be specified 

quantitatively. It is desired to compute the distance 

two random walks will proceed before joining. In this 

model each walk has the same probability that it will 

accmnulate a deviation of some given distance from 

the mean or orthogonal path. 

This would be analogous to the condition in the game 

where the house and gambler begin with the same 
THE CONCEPT OF ENTROPY ~ LANDSCAPE EVOLUTION A15 

capital, that is, Z= ~A. If the size of each wager is 

equal, then D=%A2

• 

The duration of the game under the conditions stated 

is proportional to the square of the size of the initial 

capital. However, this describes the duration of a 

gan1e with fixed boundaries. In the stream case, 

contiguous streams may join either on the right or left, 

so that the distance travelled before joining would not 

necessarily increase as the square of the separation 

distance at the last junction. On the other hand, the 

distance travelled would increase at least as the first 

power of the separation distance. Hence, there are 

two limits 

or 

Dec: A. 

In the stream case, the quantity A is equivalent to 

the average distance between streams, and we may 

assume that on the average the streams are evenly 

spaced (equivalent to the statement Z= ~A). 

Consider the geometry of a simple and regular network of joining streams. Let us follow the definition 

introduced by Horton (1945) for stream order. The 

smallest unbranched tributary is called first order, the 

strean1 receiving as tributaries only strean1s of first 

order is called second order. A stream receiving as a 

tributary a second-order stream is called third order, 

and so on. 

The geometry of joining streams may be described 

by the relation A=a2R-l where A is the mean distance 

between streams of order R, and a is the mean distance 

between the first-order streams. Hence, we can state 

that the average stream length from one junction to the 

next varies with the mean interfluve distance, in the 

form 

depending on whether the length varies as the square or 

the first power of the distance between streams of a 

given order. We deduce therefore that the logarithms 

of stream length vary linearly as the order number, and 

that the mean ratio between the lengths of streams differing in order by 1 will be between 2 and 4. This result 

denved from considerations of entropy (most probable 

state) is in agreement with the findings of Horton's 

(1945) analysis of field data on river systems. 

The mathematical relationship derived may perhaps 

best be visualized by actual trial, constructing random 

walks on cross-section paper using a table of random 

numbers. 

We constructed a stream network from the following 

elemental structure: Beginning with a series of points 

1 inch apart on a line,~ random walk was constructed, 

limited, however, only to motion right, left, or ahead. 

Backward or uphill steps were excluded. The several 

random walks when extended far enough made junctions. The resulting walk proceeding onward again 

joined and so on. A portion of the work sheet on which 

the random walks were plotted is shown as figure 6. 

To test how this constructed network compared with 

nature, we have made what is called a Horton analysis. 

This consists of plotting the logarithms of the average 

length of streams of various orders against the order 

number. Horton (1945, p. 298) found the length of 

streams of a given order increased in constant ratio with 

increasing order number. Figure 7 presents the results 

obtained from the random walks plotted, and shows that 

stream lengths increased by a ratio of about 2.8 between 

consecutive order numbers. This result lies within the 

range of 2 and 4 as suggested previously. Horton obtained, from field data he studied, ratios between 2.5 

and 3.7. Eph~meral channels near Santa Fe, N.Mex., 

were found to exhibit a ratio of about 2.1 (Leopold and 

Miller, 1956, p. 18, fig. 13). Thus the random-walk 

results give results in general quantitative agreement 

with field data. 

The constraints used in construction of the random 

walks in figure 6 may seem too limited. Another type 

of construction is as follows: Using a sheet of rectangular cross-section graph paper, each square is presumed to represent a unit area. Each square is to be 

drained, but the drainage channel from each square 

has equal chance of leading off in any of the four 

cardinal directions, subject only to the condition that, 

having made a choice, flow in the reverse direction is 

not possible. Under these conditions, it is possible for 

one or more streams to flow into a unit area, but only 

one can flow out. By the random selection of these 

directions under the conditions specified, a streom network was generated as shown on figure 8 which has 

striking similarities to natural drainage nets. Divides 

were developed, and the strea.ms joined so as to create 

rivers of increasing size. One can say that the random 

pattern represents a most probable network in a structurally and lithologically homogeneous region. The 

Horton analysis of the network pictured on figure 8 is 

presented as figure 9. 

Figure 10 shows the relation between stream length 

and drainage area. The slope of the line indicates that 

for this random-walk model length increases as the 

0.64 power of the drainage area. Simple geometric 

relations would suggest that strean1 lengths vary as the 

square root of the drainage area. This model accounts 

for the higher values that Langbein {1947, p. 135) and 

Hack (1957, p. 66-67) found for natural streams. 
A16 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

UNIFORM SPACING AT ORIGIN 

FIGURE 6.-Portion of random-walk model of stream network. 

Although in this example each elemental area is a 

square, we note that the diagram might have been 

drawn on an elastic sheet. If this sheet were stretched 

in various ways to alter the geometric figures, none of 

the relations between numbers and lengths with orders, 

shown on figure 9, would have been changed. 

Horton also showed that in natural stream networks, 

the logarithms of the number of streams of various 

orders is proportional to the order number. More 

specifically, the number of streams of a given order 

decreased in constant ratio with increasing order number. This ratio he called the "bifurication ratio." 

Figure 9 shows the relation between numbers of streams 

of the orders 1 to 5 that were developed by the streams 

on figure 8. When compared with similar diagrams 

from datn on actual rivers, given by Horton in his 

original treatment of the subject, one may note that 

in each case there is a logarithmic relationship between 
THE CONCEPT OF ENTROPY IN LANDSCAPE EVOLUTION Al7 

tOO 

en 

1-

z 

::J 

~50 

< 

0: 

1-

a; 

0: 

< 

:i 

t-

(.!) 

~ 20 ..J 

:::IE 

< L&J 

0:: 

1-

(f) 

L&J 

l!) 10 < 

0:: 

L&J 

> cC 

5 

1 2 3 4 

STREAM ORDER 

FIGURE 7.-Relation of length of streams to stream order. 

orders and numbers of streams, and moreover, the 

bifurication ratios are quite comparable. 

The essential point of these demonstrations is that 

the logarithmic increase, both of stream length and 

number of streams with order number as found iL. 

natural stream networks, aooords with the geometrical 

properties and the probabilities involved. The logarithmic relationship is thus one of optimum probability. 

Optimum probability, in this sense, represents maximum entropy. 

DISCUSSION AND GEOMORPHIC IMPLICATIONS 

In statistical mechanics the probability aspect of 

entropy has been demonstrated. The word entropy 

has, therefore, been used fu the development of systems 

other than thermodynamic ones, specifically, in information theory and in biology. Its use is continued 

here. 

With this understanding of the term, the present 

paper shows that several geomorphic forms appear to 

be explained in a general way as conditions of most 

probable distribution of energy, the basic concept in 

the term "entropy." 

It is perhaps understandable that features such as 

stream profiles which occur in nature in large numbers 

should display, on the average, conditions that might 

be expected from probability considerations because 

of the large population from which samples may be 

drawn. The difficulty in accepting this proposition 

is that there is not one but many populations owing 

to the variety of local geologic and lithologic combinations that occur. Further, the geomorphic forms seen 

in the field often are influenced by previous conditions. 

Stream channels, in particular, show in many ways the 

effects of previous climates, of orogeny, and of structural or stratigraphic relations that existed in the past. 

In some instances the present streams reflect the effects 

of sequences of beds which have been eradicated by 

erosion during the geologic past. 

In a sense, however, these conditions that presently 

control or have controlled in the past the development 

of geomorphic features now observed need not be 

viewed as preventing the application of a concept of 

maximum probability. Rather, the importance of 

these controls strengthens the usefulness and generality 

of the entropy concept. In the exa.mple presented 

here we have attempted to show that the differences 

between patterns derived from averaging random walks 

result from the constraints or controls imposed upon 

the system. We have, in effect, outlined the mathematical nature of a few of the controls which exist in 

the field. The terms in the genetic classification of 

streams reflect the operation of constraints. Terms 

like "consequent" and "subsequent" are qualitative 

statements concerning constraints imposed on streams 

which, in the absence of such constraints, would have 

a different drainage net and longitudinal profile. 

In a sensb, then, much of geomorphology has been 

the study of the very same constraints that we have 

attempted to express in a mathematical model. 

The present paper is put forward as a theoretical one. 

It is not the purpose of a theoretical paper to compare 

in detail the variety of field situations with the derived 

theoretical relations. Rather, it is hoped merely to 

provide the basis for some broad generalizations about 

the physical principles operating in the field situations. 

On the other hand, the random-walk models used 

here are simple demonstrations of how probability considerations enter into the problem. They are intended 

to exemplify how the basic equations can be tested 

experimentally. Thus the present paper should not be 

considered to deal with random walks. We hope it is 

concerned with the distribution of energy in real landscape problems. The random-walk models exemplify 

the form of the equations, but the equations describe 

the distribution of energy in real landscapes, simplified 

though the described landscape may be. 

One of the interesting characteristics of the experiments with random walks is the relatively small number 

of trials that need be carried out to obtain average 

results that closely approximate the final result from a 

large number of trials. This suggests, though it does 

not prove, that random processes operating in the landscape within certain constraints develop rather quickly 
A18 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

EXPLANATION 

1st order stream 

2nd order stream 

3rd order stream 

4th order stream 

- 5th order stream 

DRAINAGE 

.., 

Sample of squares 

showing method 

of construction of 

drainage direction 

by random choice 

FIGURE B.-Development of a random-walk drainage basin network. 
THE CONCEPT OF ENTROPY ~ LANDSCAPE EVOL~ON Al9 

the characteristics that obtain after a much longer 

period of time. 

Further, the relative insensitivity of the results to 

the lack of exactly equal probabilities among alternatives suggests that the approach to the most probable 

condition is at first very rapid. In the mathematical 

models the final elimination of minor deviations from 

the condition of maximum entropy requires a very large 

number of samples from the population of possible 

alternatives. In terms of the field development of 

forms, it seems logical that this may be equated to a 

long time-period required to eliminate minor variations 

from the theoretic most probable state. 

These observed characteristics of the averages of 

samples used in the mathematical models make it difficult to believe that in field conditions, time periods 

measured in geologic epochs could elapse before fluvial 

systems approach the condition of quasi-equilibrium. 

This line of reasoning quite fails to lend support to the 

Davisian concept that the stage of geomorphic youth 

is characterized by disequilibrium whereas the stage of 

maturity is characterized by the achievement through 

time of an equilibrium state. Rather, the reasoning 

seems to support the concept recently restated by Hack 

(1960) that no in1portant time period is necessary to 

achieve a quasi-equilibrium state. 

soo~------~------~-------r------~soo 

::t: 

..... 

(.!) 

z 

w 

..J 

::::e 

< ~------+-----~~~~---r------~10 ~ 

2 3 

STREAM OtWER 

\ 

'\ 

'\1 

4 5 

..... 

(I) 

w 

(.!) 

< 

0:: 

w 

> 

< 

FIGURE 9.-Relation of number and average lengths of streams 

to stream order for model in figure 8. 

< 

LIJ 

~ 100~-------4--~~--~---+---+--------~ 

LIJ 

(.!) 

< 

~ 50~-------;--~~+------+---+--------~ 

a: 

c 

5 10 50 100 

STREAM LENGTH. ARBITRARY UNJTS 

500 

FIGURE 10.-Relation between stream length and drainage area 

for random-walk model. 

The differing results obtained from varying the constraints imposed in the mathematical models lead us to 

following view as a working hypothesis: Landscape evolution is an evolution in the nature of constraints in time, 

maintaining meanwhile and through time essentially a 

dynamic equilibrium or quasi-equilibrium. 

This conclusion also is in agreement with the view of 

Hack (1960), though arrived at by a quite different 

line of reasoning. 

Whether or not the particular inferences stated in 

the present paper are sustained, we believe that the 

concept of entropy and the most probable state provides 

a basic mathematical conception which does deal with 

relations of time and space. Its elaboration may provide a tool by which the various philosophic pren1ises 

still characterizing geomorphology may be subjected to 

critical test. 

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preliminary announcement: U.S. Geol. Survey Circ. 421, 

23 p. 

Bell, D. A., 1956, Information theory: Sir Isaac Pitman & Sons, 

London, 17 4 p. 

Davis, W. M., 1909, The rivers and valleys of Pennsylvania: 

Geographical Essays, Dover Publications, Inc. (reprinted 

1954)' p. 413-484. 
A20 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

Denbigh, K. G., 1951, The thermodynamics of the steady state: 

Methuen & Co., London, 103 p. 

Feller, William, 1950, An introduction to probability theory and 

its applications: John Wiley & Sons, New York, 419 p. 

Hack, J. T., 1957, Studies of longitudinal stream profiles in 

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294-B, p. 46-94. 

--- 1960, Interpretation of erosional topography in humid 

temperate regions: Am. Jour. Sci., Bradley Volume, v. 

258-A, p. 80-97. 

Horton, R. E., 1945, Erosional development of streams and their 

drainage basins; hydrophysical approach to quantitative 

morphology: Geol. Soc. America Bull., v. 56, p. 275-370. 

Khinchin, A. I., 1957, Mathematical foundations of information 

theory: Dover Publications, Inc., New York, 120 p. 

Langbein, W. B., 1947, Topographic characteristics of drainage 

basins: U.S. Geol. Survey Water-Supply Paper 968-C, 

p. 125-155. 

Laursen, E. M., 1958, Sediment---transport mechanics In stable 

channel design: Am. Soc. Civil Engrs. Trans. v. 123, p. 

19~206. 

Leopold, L. B., 1953, Downstream change of velocity in rivers: 

Am. Jour. Sci., v. 251, p. 606-624. 

Leopold, L. B., and Maddock, T., Jr., 1953, The hydraulic 

geometry of stream channels and some physiographic 

implications: U.S. Geol. Survey Prof. Pape~ 252, 56. P• 

Leopold, L. B., and Miller, J. P., 1956, Ephemeral streamsHydraulic factors and their relation to the drainage net: 

U.S. Geol. Survey Prof. Paper 282-A, p. 1-36. 

Lewis, G. N., and Randall, Merle, 1961, Thermodynamics: 

McGraw-Hill Book Co., Inc., 723 p. 

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Hardin and Brussels quadrangles: U.S. Geol. Survey Prof. 

Paper 217, 175 p. 

*U.S. GOVERNMENT PRINTING OFFICE: 1967 0- 275-097