ENTROPY AND“THERMODYNAMIC” RELATIONSFOR NONEQUILIBRIUM STEADYSTATES
HAL TASAKIWITH T.S.KOMATSU, N.NAKAGAWA, S.SASA
PRL 100, 230602 (2008), arXiv:0711.0246
J. STAT. PHYS. 159, 1237 (2015), arXiv:1405.0697

webinar, November 2020

TWO “TWISTS” IN
STEADY STATE THERMODYNAMICSHAL TASAKIWITH T.S.KOMATSU, N.NAKAGAWA, S.SASA
PRL 100, 230602 (2008), arXiv:0711.0246
J. STAT. PHYS. 159, 1237 (2015), arXiv:1405.0697

webinar, November 2020

MOTIVATION

sha1_base64="ny+fJALrqBCxkat3RzRGGU8N8rY=">AAACCXicbVA9SwNBEJ2LXzF+nVraHAbBQsOdKNoIQRvLCPmC5Ax7m71kyd7usbsnhCOtjb/DzsZCEVv/gZ3/xk1yhSY+GHi8N8PMvCBmVGnX/bZyC4tLyyv51cLa+sbmlr29U1cikZjUsGBCNgOkCKOc1DTVjDRjSVAUMNIIBtdjv3FPpKKCV/UwJn6EepyGFCNtJGHb8AQBENCA4BKqcAcpHIMHIzgyDoekYxfdkjuBM0+8jBQhQ6Vjf7W7AicR4RozpFTLc2Ptp0hqihkZFdqJIjHCA9QjLUM5iojy08knI+fAKF0nFNIU185E/T2RokipYRSYzgjpvpr1xuJ/XivR4YWfUh4nmnA8XRQmzNHCGcfidKkkWLOhIQhLam51cB9JhLUJr2BC8GZfnif1k5J3VnJvT4vlqyyOPOzBPhyaQM+hDDdQgRpgeIBneIU369F6sd6tj2lrzspmduEPrM8f9rmUKQ==</latexit><latexit

EQUILIBRIUM THERMODYNAMICS

PHYSICAL SYSTEM WITH CONTROLLABLE PARAMETERS
1START FROM THE EQUILIBRIUM WITH = T , ⌫ E.G., THE
0 0
CHANGE THE PARAMETERS TO , ⌫VOLUME
EQUILIBRIUM

S( , ⌫ )00EQUILIBRIUMSUDDEN CHANGERELAXATION TO NEW EQUILIBRIUM

SYSTEM DURING THE RELAXATION PROCESS
CLAUSIUS RELATIONQ ENERGY (HEAT) TRANSFERRED TO THE BATHS FROM THE

S( , ⌫) =Q + O(( Q) )2STARTING POINT OF EQUILIBRIUM THERMODYNAMICSSETNONEQUILIBRIUM
STEADY STATE (NESS)

1 6=2 , AND FIX

121,2, ⌫12AS t " 1 THE SYSTEM IS EXPECTED TO APPROACH A UNIQUESTATIONARY STATE = NONEQUILIBRIUM STEADY STATE (NESS)(PROVIDED THAT THE “DEGREE OF NONEQULIBRIUM” IS SMALL)
EQUILIBRIUM STATE:NESS:NO MACROSCOPIC CHANGESNO MACROSCOPIC FLOWSNO MACROSCOPIC CHANGESNONVANISHING MACROSCOPICFLOW OF ENERGY OR MATTEROPERATION TO NESSTHERMODYNAMICS FOR NESS?
START FROM THE NESS WITH 1 , 2 , ⌫000CHANGE THE PARAMETERS TO 1 , 2 , ⌫NESS121SUDDEN CHANGE2NESS1RELAXATION TO NEW NESSIS THERE ANYTHING LIKE THE CLAUSIUS RELATION?
DERIVATION BASED ON GENERALMICROSCOPIC MODELS

2TYPICAL SYSTEMSYSTEM OF PARTICLESCLASSICAL MECHANICAL SYSTEM WITH N PARTICLES IN A
FINITE BOX ⇤r i 2 ⇤ ⇢ R POSITION
OF THE i-TH PARTICLE
3pi 2 R MOMENTUM
p = mv3ix = (r 1 , . . . , r N ; p1 , . . . , pN )⇤T1T2THE SYSTEM IS ATTACHED TO TWO HEAT BATHSi

TIME EVOLUTIONUSUAL NEWTON EQUATIONd r i (t)m=2dt2gradi V (r 1 (t), . . . , r N (t))( )Vext (r i ) +V (r 1 , . . . , r N ) =iv(r irj )i<j(CONTROLLABLE) PARAMETER (E.G., THE VOLUME)
MARKOVIAN TIME EVOLUTION AT THE WALLS⇤THERMAL WALL,LANGEVIN DYNAMICS
T1 (ONLY NEAR THE WALLS),T2ETC.
WE NEED LOCAL DETAILED BALANCE CONDITION

vTv
sha1_base64="YueQNemtCkHA85SzJlKLQnNjWjY=">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</latexit>

vout<latexit

A PARTICLE WITH ANY INCIDENT VELOCITY v IS BOUNCED BACKout
WITH A RANDOM VELOCITYv
WITH THE PROBABILITY DENSITYTHERMAL WALLinout 2 ⇤⇥m |v |outoutpT (v ) = A vx exp2kT

out out out= (vx , vy , vz )inoutxoutvx > 0m in 2q = |v |21 ⇣ m ⌘2A=2⇡ kTk BOLTZMANN CONSTANTT TEMPERATURE OF THE WALL

out outvy , vz 2 R

ENERGY (HEAT) TRANSFERED FROM THE
SYSTEM TO THE BATHm out 2|v |2

GENERAL SETUP

BASIC INGREDIENTSCONTROLLABLE PARAMETERS= (⇥1 , ⇥2 , ⇤)(INVERSE) TEMPERATURES
OF THE BATHS

12MODEL PARAMETERS⌫⌫
1x STATE OF THE SYSTEMx $ (r 1 , . . . , r N ; p1 , . . . , pN )⇤x $ (r 1 , . . . , r N ; p1 , . . . , pN )⌫⌫ENERGY Hx = Hx⇤

2

TIME EVOLUTION
TIME INTERVAL t[0, ]OPERATION BY OUTSIDE AGENT

FIXED PROTOCOL (OR FUNCTION)(t) = (⇥1 (t), ⇥2 (t), ⇤(t))ˆ = ( (t))t [0, ]PATH x̂ = (x(t))t2[0,⌧ ]MARKOV DYNAMICS WITH PATH PROBABILITY DENSITY TZ↵ˆDx̂ T [x̂] = 1

T[x̂ ] = e†ˆ⇥↵[x̂][x̂]x(0)=xinit

DETAILED FLUCTUATION THEOREM↵ˆ†↵ˆ

T [x̂]↵ˆˆ = (↵(⌧
TIME REVERSED PROTOCOL ↵†⇤TIME REVERSED PATHx̂ = (x (⌧†t))t2[0,⌧ ]t))t2[0,⌧ ]ENTROPY PRODUCTION
DETAILED FLUCTUATION THEOREMT†↵ˆ↵ˆ[x̂ ] = e†⇥ [x̂]

T [x̂]↵ˆ⇥ [x̂] TOTAL ENTROPY PRODUCTION (IN THE BATHS)↵ˆWHEN THE TEMPERATURES ARE KEPT CONSTANT⇥ [x̂] =↵ˆ

IN GENERAL⇥ [x̂] =↵ˆ

1 Qsys!1 [x̂] +Z ⌧0jsys!1dtX

2 Qsys!2 [x̂]i (t) jsys!i [x̂](t)i=1,2jsys!2ENTROPY PRODUCTION RATE
IN THE i-TH BATHNESS AND THE AVERAGE
WHEN THE PARAMETERS ARE KEPT CONSTANTTHE SYSTEM EVENTUALLY CONVERGES TO A UNIQUENONEQUILIBRIUM STEADY STATE (NESS)↵⇢x PROBABILITY DISTRIBUTION OF NESS WITH ↵PATH AVERAGE OF ANY FUNCTION F [x̂]hF i :=↵ˆZ↵(0) ↵ˆDx̂ ⇢x(0) T [x̂] F [x̂]START FROM THE NESS FOR THE INITIAL PARAMETERS
ˆAND CHANGE THE PARAMETER ACCORDING TO ↵CLAUSIUS RELATIONAND EXTENDED CLAUSIUSRELATIONEQUILIBRIUM CASEOPERATION BETWEEN TWO EQUILIBRIUM STATES(⇥, ⇥, ⇤),t [0, ⌅ /2](t) =(⇥ , ⇥ , ⇤ ), t (⌅ /2, ⌅ ]AMOUNT OF CHANGE ⇥ = max{|EQUILIBRIUM|, |⇤⇤|}

EQUILIBRIUM

STANDARD CLAUSIUS RELATION (FOR LARGE ⌧ )S( ,,⇤ )S( , , ⇤) =⇥⇤ + O(⇥ )ˆ ˆ

THERMODYNAMIC ENTROPY = SHANNON ENTROPY OF ⇢S(↵) =R↵↵dx⇢x log ⇢x2↵THE MEANING OF THE
CLAUSIUS RELATIONSTANDARD CLAUSIUS RELATION

S( ,,⇤ )S( , , ⇤) =⇥⇤ + O(⇥ )ˆ ˆ2fin
SbathsS( ,init, ⇤) + Sbaths = S(,initSbaths
fin2, ⇤ ) + Sbaths + O(⇥ )THE TOTAL ENTROPY OF {THE SYSTEM + THE BATHS} ISCONSTANT
THIS IS NO LONGER TRUE IN NESS!

THERE IS A CONSTANT
ENTROPY PRODUCTION

ENTROPY PRODUCTION IN NESSENTROPY PRODUCTION RATE IN NESS
↵st =↵
1 jst / ( 2↵2 jst↵STATIONARY CURRENT jst /21)

21jst

12

↵
st = 0↵st > 01 =21 =2NESS IS ACCOMPANIED BY A CONSTANT NONVANISHINGENTROPY PRODUCTION IN THE BATHSCLAUSIUS RELATION FOR NESS?
OPERATION BETWEEN TWO NESS(t) =[0, ⌅ /2](⌅ /2, ⌅ ](⇥1 , ⇥2 , ⇤), t(⇥1 , ⇥2 , ⇤ ), t

NESS
1NESS
2IS IT POSSIBLE THAT S( 1 ,NO! BECAUSE01,002, ⌫ )122, ⇥ )S( 1 ,12 , ⇥) ⇥2⇤ˆ ˆ ?

S( 1 , 2 , ⌫) IS INDEPENDENT OF ⌧⌧ ↵ ⌧ ↵0↵ˆ ↵ˆ
h⇥ i ⇠st +st DIVERGES AS ⌧ " 122S(⌅ENTROPY PRODUCTIONENTROPY PRODUCTION RATE(t) =OPERATION IN
EQUILIBRIUMOPERATION INNESS(t)(t)↵st↵0st1 jsys!1 (t) +2 jsys!2 (t)TOTAL ENTROPY PRODUCTIONR⌧dt0h⇥ i =↵ˆ ↵ˆ(t)t

TOTAL ENTROPY PRODUCTION
IS DIVERGENT IN TIMEh⇥ i =↵ˆ ↵ˆR⌧dt(t)0tEXCESS ENTROPY PRODUCTIONOONO-PANICONITOTAL ENTROPY PRODUCTION
IS DIVERGENT IN TIMEEXCESS ENTROPY(t)↵ˆ ↵ˆh⇥ i
PRODUCTION

OPERATION INNESS↵ˆ ↵ˆh⇥ex i“RENORMALIZED”
FINITE QUANTITY:EXCESS ENTROPYPRODUCTION↵ˆ↵ˆ⇥ex [x̂] := ⇥ [x̂]tZ ⌧“BARE” ENTROPYPRODUCTION0dt↵(t)st“HOUSE KEEPING”
ENTROPY PRODUCTION

EXTENDED CLAUSIUS RELATIONOPERATION BETWEEN TWO NESS(t) =(⇥1 , ⇥2 , ⇤), t(⇥1 , ⇥2 , ⇤ ), tNESS
1[0, ⌅ /2](⌅ /2, ⌅ ]NESS21212THERE EXISTS ENTROPY OF NESS, AND WE HAVE

S(01,002, ⌫ )S( 1 , 2 , ⌫)↵ˆ ↵ˆ
22= h⇥ex i + O(✏ ) + O( )AMOUNT OF CHANGE ⇥ = max{| 1DEGREE OF NONEQUILIBRIUM

1 |, | 2⇥ = max{| 12 |, |⇤2 |, | 1⇤|}2 |}MICROSCOPIC EXPRESSION
FOR THE ENTROPY
THE NONEQUILIBRIUM ENTROPY IS RELATED TO THE PROBABILITYDENSITY BYS(↵) = Ssym [⇢ ]↵WITH THE “SYMMETRIZED SHANNON ENTROPY”

Ssym [⇢] :=Zdx ⇢x logp⇢x ⇢x⇤STATE x $ (r 1 , . . . , r N ; p1 , . . . , pN )⇤TIME REVERSAL x$ (r 1 , . . . , r N ; p1 , . . . , pN )

THE FIRST “TWIST” IN SST
THE NONEQUILIBRIUM ENTROPY IS RELATED TO THE PROBABILITYDENSITY BYS(↵) = Ssym [⇢ ]↵WITH THE “SYMMETRIZED SHANNON ENTROPY”

Ssym [⇢] :=Zdx ⇢x logp⇢x ⇢x⇤STATE x $ (r 1 , . . . , r N ; p1 , . . . , pN )⇤TIME REVERSAL x$ (r 1 , . . . , r N ; p1 , . . . , pN )

ADIABATIC LIMIT
FOR “SLOW AND GENTLE” PROTOCOL ˆ = ((t))t [0, ]WITH ↵(0) = ( 1 , 2 , ⌫) AND ↵(⌧ ) = (S( 1 ,2, ⌅ )S( 1 ,2 , ⌅) =AMOUNT OF CHANGE ⇥ = max{| 1DEGREE OF NONEQUILIBRIUM⇥01,002, ⌫ )ˆ ˆ2ex ⇤ + O(⇤ ⇥)1 |, | 2⇥ = max{| 12 |, |⇤2 |, | 1⇤|}2 |}A “NATURAL” EXTENSION OF THE TRADITIONAL CLAUSIUS
RELATION, IN WHICH (DIVERGENT) “BARE” ENTROPY PRODUCTIONIS REPLACED BY ITS “RENORMALIZED” COUNTERPARTMEANINGFUL WHEN THE DEGREE OF NONEQULIBRIUM IS SMALL

OPERATIONAL
DETERMINATION OFENTROPY

OPERATION BETWEENEQULIBRIUM AND NESSˆ PROTOCOL WHICH BRINGS( 1 ,1 , ⇥) TO ( 1 ,2 , ⇥)BY CHANGING ONLY THE TEMPERATURE OF THE BATH 2.
EQUILIBRIUMNESS
11EQUILIBRIUM ENTROPYS( 1 ,2 , ⌅)S( 1 ,1⇥=
1 , ⌅) =221 =⇤O( )ˆ ˆ2⇥ ex ⇤ + O(⇤ ⇥)3WE CAN DETERMINE THE NONEQUILIBRIUM ENTROPY TO
2O( ) ONLY BY MEASURING THE HEAT CURRENTS!

COMPARING ENTROPIESOF TWO NESSSUPPOSE THAT WE WANT TO DETERMINE THE DIFFERENCE

S( 1 ,2, ⇥ )S( 1 ,TEMPERATURE
OF THE RIGHT BATH
22 , ⇥)TEMPERATURES ARE
THE SAME|| = O(1)1PARAMETERDIRECT PATH
FIX THE TEMPERATURES AND CHANGENESS
12TONESS
21O( )
ˆ ˆ2⇥ ex ⇤ + O(⇤ ⇥)FROM THE EXTENDED CLAUSIUS RELATION, ONE GETS

S( 1 ,2, ⌅ )= |⇥S( 1 ,2 , ⌅) =⇥| = O(1)WE CAN DETERMINE THE DIFFERENCE
ONLY WITH THE PRECISION OF O( )22INDIRECT PATHUSE THE COMBINATION OF THE THREE PROCESSES( 1,2 , ⇥)a( 1,1 , ⇥)b( 1,1c,⇥ ) (1,2, ⇥ )FROM THE EXTENDED CLAUSIUS RELATION, ONE GETS

S( 1 ,2, ⇥ )=S( 1 ,h⇥ex ia

WE CAN DETERMINE THE
DIFFERENCE WITH THE
2PRECISION OF O( )2 , ⇥)
1Qbh⇥ex ic + O(✏ )3

TEMPERATURE
OF THE RIGHT BATH
21EQUILIBRIUM

1PARAMETERPOSSIBLE ERROR IN EACHPROCESSTEMPERATURE
OF THE RIGHT BATH

O( )22O( )CAN’T WE DO
BETTER?

31O( )
30EQUILIBRIUM

1PARAMETERTHE TEMPERATURE OF THE LEFT BATH IS FIXED AT

1NONLINEARNONEQULIBRIUM
RELATION

THE SECOND ORDER EXTENDED
CLAUSIUS RELATIONFOR THE DIRECT PATH FROM ( 1 , 2 , ⇥) TO ( 1 , 2 , ⇥ )NESSNESS
1S( 1 ,=2212, ⇥ )S( 1 , 2 , ⇥)1+ 2ˆ ˆˆ⇥W ;⇥ ex ⇤ +4W WORK DONE TO THE SYSTEM⇥W ; ⇤ := ⇥W⇤⇤ + O(⇤ ⇥)ˆ ˆ= |⇥⇥W ⇤⇥ ⇤3⇥|

THE SECOND ORDER EXTENDED
CLAUSIUS RELATION
S( 1 ,=2, ⇥ )S( 1 , 2 , ⇥)1+ 2ˆ ˆˆ⇥W ;⇥ ex ⇤ +4⇤ + O(⇤ ⇥)ˆ ˆ3THE RELATION TAKES INTO ACCOUNT “NONLINEARNONEQUILIBRIUM” CONTRIBUTIONS, AND HAS A DESIRED HIGHERPRECISION.
BUT IT CONTAINS A CORRELATION BETWEEN HEAT AND WORK.
IT IS A RELATION BETWEEN MACROSCOPIC QUANTITIES; BUT CANWE CALL IT A THERMODYNAMIC RELATION?

OPERATIONAL
DETERMINATION OF ENTROPYDETERMINE S( 1 ,2, ⇥ )S'2S'1S( 1 ,h⇥ex i +h⇥ex i

S=2 , ⇥) TO THE ORDER O(1+24S'1hW ; ⇥ih⇥ex i

EQUILIBRIUM

Q

1ONE HAS TO USE EITHER THE 1ST ORDER RELATIONS OR THE 2ND
ORDER RELATION, DEPENDING ON THE PATHS!

2)

THE SECOND “TWIST” IN SSTDETERMINE S( 1 ,2, ⇥ )S'2S'1S( 1 ,h⇥ex i +h⇥ex i

S=2 , ⇥) TO THE ORDER O(1+24S'1hW ; ⇥ih⇥ex i

EQUILIBRIUM

Q

1ONE HAS TO USE EITHER THE 1ST ORDER RELATIONS OR THE 2NDORDER RELATION, DEPENDING ON THE PATHS!

2)

SUMMARYOUR RESULTS ARE MATHEMATICALLY RIGOROUS
FOR MARKOV JUMP PROCESSES, BUT NOTENTIRELY RIGOROUS FOR OTHER MODELSWE FOUND A NATURAL EXTENSION OF CLAUSIUS
RELATION FOR OPERATIONS BETWEEEN NESS,WHICH ENABLES ONE TO OPERATIONALLY
DETERMINE NONEQULIBRIUM ENTROPY TO THESECOND ORDER IN ⇥ = | 12|THE NONEQUILIBRIUM ENTROPY HAS ANEXPRESSION IN TERMS OF SYMMETRIZEDSHANNON ENTROPY RpSsym [⇢] :=dx ⇢x log ⇢x ⇢x⇤
WHAT DOES IT MEAN??

IS THERE A MEANINGFUL THERMODYNAMICS
FOR NESS WHICH YIELDS NONTRIVIAL
EXPERIMENTAL PREDICTIONS?
THERE ARE MANY DIFFERENT ATTEMPTS, E.G., BY
Netochny-Maes, Jona-Lasinio et al., NakagawaSasa, ….