unattainability of carnot efficiency in the brownian heat engine we discuss the reversibility of the brownian heat engine we perform an asymptotic analysis of the kramers equation on a buttiker landauer system and show quantitatively that carnot efficiency is unattainable even in the fully overdamped limit the unattainability is attributed to inevitable irreversible heat flow over the temperature boundary

covering by random intervals and one dimensional continuum percolation a brief historical introduction is given to the problem of covering a line by random overlapping intervals the problem for equal intervals was first solved by whitworth in the 1890s a brief resume is given of his solution the advantages of the present author s approach which uses a poisson process are outlined and a solution is derived by laplace transforms the method of hammersley for dealing with a stochastic distribution of intervals is described and a solution can still be derived by laplace transforms the asymptotic behavior as the line becomes long is calculated and is related to the one dimensional continuum percolation problem it is shown that as long as the mean interval size is finite the probability of complete coverage decays exponentially so that the critical percolation probabilityp c 1 however as soon as the mean interval size becomes infinite the critical percolation probabilityp c switches to 0 this is in accord with previous results for a lattice model by chinese workers but differs from those of schulman a possible reason for the discrepancy is a difference in boundary conditions

fokker planck equations for time delayed systems via markovian embedding for stochastic systems with discrete time delay the fokker planck equation fpe of the one time probability density function pdf does not provide a complete self contained probabilistic description it explicitly involves the two time pdf and represents in fact only the first member of an infinite hierarchy we here introduce a new approach to obtain a fokker planck description by using a markovian embedding technique and a subsequent limiting procedure on this way we find a closed complete fpe in an infinite dimensional space from which one can derive a hierarchy of fpes while the first member is the well known fpe for the one time pdf we obtain as second member a new representation of the equation for the two time pdf from a conceptual point of view our approach is simpler than earlier derivations and it yields interesting insight into both the physical meaning and the mathematical structure of delayed processes we further propose an approximation for the two time pdf which is a central quantity in the description of these non markovian systems as it directly gives the correlation between the present and the delayed state application to a prototypical bistable system reveals that this approximation captures the non trivial effects induced by the delay remarkably well despite its surprisingly simple form moreover it outperforms earlier approaches for the one time pdf in the regime of large delays

nonequilibrium thermodynamics of stochastic systems with odd and even variables the total entropy production of stochastic systems can be divided into three quantities the first corresponds to the excess heat while the second two comprise the housekeeping heat we denote these two components the transient and generalized housekeeping heat and we obtain an integral fluctuation theorem for the latter valid for all markovian stochastic dynamics a previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal which restricts consideration of directional variables such as velocity

percolation and contact processes with low dimensional inhomogeneity we consider inhomogeneous nearest neighbor bernoulli bond percolation on z d where the bonds in a fixed s dimensional hyperplane 1 s d 1 have density p 1 and all other bonds have fixed density p c z d the homogeneous percolation critical value for s 2 it is natural to conjecture that there is a new critical value p c s z d for p 1 strictly between p c z d and p c z s we prove this for large d and 2 s d 3 for s 1 it is natural to conjecture that p c 1 z d 1 as shown for d 2 by zhang we prove this for large d related results for the contact process are also presented

critical values in a long range percolationon spaces like fractals on a certain class of general discrete spaces including fractals we consider a model in which each pair of distinct points is connected by a random bond the main question we are concerned is whether a connected component consisting of infinitely many points exists or not this depends on the choice of parameters in the connecting probabilities and the aim of this paper is to find thresholds of the parameters

simple random walk on long range percolation clusters ii scaling limits supported in part at the technion by a landau fellowship supported in part by an alfred sloan fellowship in mathematics

role of sufficient statistics in stochastic thermodynamics and its implication to sensory adaptation a sufficient statistic is a significant concept in statistics which means a probability variable that has sufficient information required for an inference task we investigate the roles of sufficient statistics and related quantities in stochastic thermodynamics specifically we prove that for general continuous time bipartite networks the existence of a sufficient statistic implies that an informational quantity called the sensory capacity takes the maximum since the maximal sensory capacity imposes a constraint that the energetic efficiency cannot exceed one half our result implies that the existence of a sufficient statistic is inevitably accompanied by energetic dissipation we also show that in a particular parameter region of linear langevin systems there exists the optimal noise intensity at which the sensory capacity the information thermodynamic efficiency and the total entropy production are optimized at the same time we apply our general result to a model of sensory adaptation of e coli and find that the sensory capacity is nearly maximal with experimentally realistic parameters

further comments on the one dimensional percolation problem with multineighbour bonds one dimensional site and bond percolation problems with bonds connecting lth nearest neighbours are solved exactly by using an infinite cell to cell renormalisation group transformation both thermal and magnetic critical exponents are obtained through the scaling relations although universality breaks down for site and bond percolation when l or 2 suzuki s weaker universality 1974 still holds

directed percolation shape of the percolation cone conductivity exponents and high dimensionality behaviour presents several results for the directed percolation problem the shape of the percolation cone is discussed showing how the operating angle behaves near threshold and explaining the orientation dependence of the correlation exponents found by domany and kinzel 1981 the cayley tree model is used to discuss the high dimensionality limit the conductivity problem on a directed network is introduced and the high dimensionality values for the conductivity exponents s and tau are found

the growth of the infinite long range percolation cluster we consider long range percolation on z d where the probability that two vertices at distance r are connected by an edge is given by p r 1 exp lambda r is an element of 0 1 and the presence or absence of different edges are independent here lambda r is a strictly positive nonincreasing regularly varying function we investigate the asymptotic growth of the size of the k ball around the origin vertical bar b k vertical bar that is the number of vertices that ale within graphdistance k of the origin for k infinity for different lambda r we show that conditioned on the origin being in the unique infinite cluster nonempty classes of nonmereasimi regularly varying lambda r exist for which respectively vertical bar b k vertical bar 1 k infinity almost surely there exist 1 infinity p a 1 1 almost surely this result can be applied to spatial sir epidemics in particular regimes are identified for which the basic reproduction number r 0 which is an important quantity for epidemics in unstructured populations has a useful counterpart in spatial epidemics

random fixed points and viable random solutions of functional differential inclusions abstract there are given conditions under which the existence of viable solutions yields the existence of viable random solutions to functional differential inclusions

caratheodory type selections and random fixed point theorems abstract we provide some new caratheodory type selection theorems i e selections for correspondences of two variables which are continuous with respect to one variable and measurable with respect to the other these results generalize simultaneously michael s 21 continuous selection theorem for lower semicontinuous correspondences as well as a caratheodory type selection theorem of fryszkowski 10 random fixed point theorems which generalize ordinary fixed point theorems e g browder s 6 follow as easy corollaries of our results

the diameter of a long range percolation cluster on generalized pre sierpinski carpet and regular tree we consider some kinds of graphs obtained by generalizing the pre sierpinski carpet which is one of well known fractal lattices here we deal with not only some fractal lattices but also many irregular graphs which do not belong to fractal lattices under several assumptions we present estimates on a diameter of the long range percolation cluster on such graphs in misumi j stat phys 158 1083 1089 2015 estimates on the graph diameter of the long range percolation cluster are established on the pre sierpinski gasket which is another fundamental fractal lattice and the result of this paper is its extension in a certain sense we will also discuss the behavior of the diameter of the long range percolation graph on a regular tree which seems to be quite different from the result on the generalized pre sierpinski carpet

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

caratheodory selections for multivalued mappings abstract we prove the existence of a caratheodory selection for a set valued mapping satisfying standard conditions also an application to random fixed point for random operators is established

series expansion studies of directed percolation estimates of the correlation length exponents series expansion methods have been used to estimate the longitudinal and transverse correlation length exponents for directed bond percolation on the square lattice the central estimate for the ratio of these exponents phi 1 579 or 0 006 is in good agreement with the value obtained by the phenomenological scaling renormalisation group method

scaling of overhangs in 1 1 dimensional directed processes in a gradient when a gradient is imposed in the control parameter of a directed process in a 1 1 dimensional directed process a front may occur the position of this front evolves in steps of size one in the direction opposite to the gradient and in steps of size equal to or larger than one in the other direction the larger steps are due to overhangs there is a critical value for the control parameter only if the first moment of the step size distribution in the direction of the gradient is not singular i e it takes on a finite value in the thermodynamic limit if there is a critical point this step size distribution is a power law at this point with an exponent larger than two and which the authors conjecture to be equal to 3 beta nu perpendicular to beta is the order parameter exponent and nu perpendicular to is the spatial correlation length exponent in a finite gradient there is a second exponent governing the upper cut off in the step size distribution this exponent is equal to nu perpendicular to 1 nu perpendicular to these two exponents govern the convergence of the position of the front towards the critical value of the control parameter the authors exemplify their discussion with directed site percolation and with two versions of a model originally designed to study catalytic reactions

one dimensional 1 j i s percolation models the existence of a transition for s 2 consider a one dimensional independent bond percolation model withpj denoting the probability of an occupied bond between integer sitesi andi j j 1 ifpj is fixed forj 2 and mathop lim limits j to infty j2pj 1 then unoriented percolation occurs forp1 sufficiently close to 1 this result analogous to the existence of spontaneous magnetization in long range one dimensional ising models is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size oriented percolation is shown to occur forp1 close to 1 if mathop lim limits j to infty jspj 0 for somes 2 analogous results are valid for one dimensional site bond percolation models

multiple phase transitions in long range first passage percolation on square lattices we consider a model of long range first passage percolation on the d dimensional square lattice in which any two distinct vertices are connected by an edge having exponentially distributed passage time with mean where 0 is a fixed parameter and is the l1 norm on we analyze the asymptotic growth rate of the set t which consists of all such that the first passage time between the origin 0 and x is at most t as t we show that depending on the values of there are four growth regimes i instantaneous growth for 2d 1 like the nearest neighbor first passage percolation model corresponding to 2015 wiley periodicals inc

percolation properties of a three dimensional random resistor diode network studies percolation in a random resistor diode network on the simple cubic lattice here the occupied bonds joining nearest neighbour sites may be either resistor like transmitting information or connectivity in either direction along a bond or diode like transmitting in only one direction they consider a model in which there exist both positive diodes which transmit in the x y or z directions and negative diodes which transmit oppositely they apply positive space renormalisation group methods to map out the phase diagram and calculate the exponents associated with the phase transitions of this system they find three novel types of transitions due to the presence of the diodes with only one species of diodes and vacancies unoccupied bonds present a directed threshold occurs at a critical concentration of diodes here an infinite cluster forms which transmits only in the direction of the average diode polarisation a reverse threshold occurs when resistors and only one species of diodes exist at this point the resistors mediate information flow in the infinite cluster opposite to the diode polarisation finally with all bond elements present a mixed threshold occurs

directed percolation a finite size renormalisation group approach the finite size renormalisation group technique introduced by nightingale 1976 is applied to the directed percolation problem the decay of correlations is anisotropic in this model and finite size scaling is extended to treat such anisotropy precise estimates for critical exponents and percolation probabilities are obtained for site bond and site bond percolation on the square lattice with bonds directed along the positive axes both free boundary conditions for which the results converge linearly with 1 n as n to infinity and helical boundary conditions for which unexpectedly the results converge linearly with 1 n3 are considered

application of logical transport to determine the directed and isotropic percolation thresholds the authors present a monte carlo algorithm based on logical transport which determines the effective isotropic percolation and directed percolation thresholds of finite size lattices they apply the algorithm to bond and site directed percolation in two dimensions and determine the asymptotic percolation thresholds by finite size scaling

transience recurrence and critical behavior for long range percolation we study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions d 1 2 where x and y are connected with probability we show that if d s 2d then there is no infinite cluster at criticality this result is extended to the free random cluster model a second corollary is that when d 2 and d s 2d we can erase all long enough bonds and still have an infinite cluster the proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks in particular we show that i i d conductances on a recurrent graph of bounded degree yield a recurrent electrical network

series expansion analysis of directed site bond percolation on the square and simple cubic lattices analysis of low density series for site bond percolation on the directed square sq and simple cubic sc lattice and related series for bond percolation on the honeycomb h and diamond di lattices is found to be consistent with pc sq site bond pc h bond 0 8228 or 0 0002 pc sc site bond pc di bond 0 637 or 0 002 and previous estimates of gamma nu sub and nu perpendicular to analysis of the square lattice series supports the validity of the scaling relation gamma 0 gamma d 1 nu perpendicular to for the two dimensional lattices site percolation on the honeycomb and diamond lattices is also considered

long range percolation in one dimension the problem of long range percolation in one dimension is proposed the authors consider a one dimensional bond percolation system with bonds connecting an infinite number of neighbours where the occupation probability for the nth nearest neighbour bond pn varies as p1 ns using the transfer matrix method they find that when s 2 only the short range percolation exists namely the system percolates only when p1 1 a transition to long range percolation is found at s 2 where the percolation threshold drops suddenly from the short range value p1c 1 to the long range value p1c 0

integral fluctuation theorem for the housekeeping heat the housekeeping heat qhk is the dissipated heat necessary to maintain the violation of detailed balance in nonequilibrium steady states by analysing the evolution of its probability distribution we prove an integral fluctuation theorem exp qhk 1 valid for arbitrary driven transitions between steady states we discuss gaussian limiting cases and the difference between the new theorem and both the hatano sasa and the jarzynski relation

decay properties of the connectivity for mixed long range percolation models on z d in this short note we consider mixed short long range independent bond percolation models on k d let puv be the probability that the edge u v will be open allowing a x y dependent length scale and using a multi scale analysis due to aizenman and newman we show that the long distance behavior of the connectivity xy is governed by the probability pxy the result holds up to the critical point

entropy production and fluctuation theorems for langevin processes under continuous non markovian feedback control continuous feedback control of langevin processes may be non markovian due to a time lag between the measurement and the control action we show that this requires one to modify the basic relation between dissipation and time reversal and to include a contribution arising from the noncausal character of the reverse process we then propose a new definition of the quantity measuring the irreversibility of a path in a nonequilibrium stationary state which can also be regarded as the trajectory dependent total entropy production this leads to an extension of the second law which takes a simple form in the long time limit as an illustration we apply the general approach to linear systems that are both analytically tractable and experimentally relevant

the diameter of a long range percolation cluster on pre sierpinski gasket we give several bounds for a graph diameter of a random graph created by a stochastic model called the long range percolation in which any pair of two distinct points is connected by a random bond independently such a problem is well studied on a finite subset of the d dimensional square lattice where d is a positive integer in this manuscript we consider the problem on a finite subset of a fractal lattice which is called the pre sierpinski gasket we can observe that the hausdorff dimension of the fractal lattice appears in the critical parameters of s where s is a value determining the order of probabilities that random bonds exist

a long range domany kinzel model of directed percolation a long range domany kinzel 1981 model proposed by wu and stanley 1982 is solved using random walk formulations in this model for every site i j in a two dimensional lattice there is a directed bond present from site i j to i 1 j with probability one there are also m 1 directed bonds present from i j to i k j 1 k 1 0 1 2 3 m 1 with respective probabilities pk 1 where m is any positive integer an exact expression is obtained to determine the critical percolation angle theta c for any distribution of pn the system percolates in the region theta c theta or 0 with probability one and zero outside this region where theta is the angle measured from the x axis if the authors let m go to infinity and pn varies as p1n s they find that when s or approximately 2 when m is large but finite theta c is also obtained for the following two distributions a pn a a n with a 0 b pn beta m and beta 0

directed percolation mean field theory and series expansions for some two dimensional lattices the shape of percolation clusters in the directed percolation problem is studied within mean field theory low density series expansions are obtained for bond and site problems on the square and triangular lattices these are used to test a scaling formula for the pair connectedness which involves two lengths xi sub rho and xi perpendicular to rho determination of the exponents nu sub and nu perpendicular to for these lengths shows that all four problems are in the same universality class and the values support the hyperscaling relation beta 1 2 d nu perpendicular to nu sub gamma an independent argument is given for this relation

random graphs and network communication the problem of random graphs which arises in the analysis of network reliability in communication theory is considered here as a bond percolation a closed form expression is obtained for the cluster size generating function from which the mean cluster size as well as the percolation probability are derived in a network of n to infinity stations in which the communication between any two stations is intact with a probability alpha n it is found that for alpha 1 there is a non zero percolation probability

comments on the transfer matrix approach to the one dimensional bond percolation with further neighbour bonds the transfer matrices which have been used in previous studies on the one dimensional bond percolation problem with bonds connecting lth nearest neighbours are only for the case of directed bonds for the case of undirected bonds the transfer matrices are non systematic and become much more complicated when l or 3 here the authors study the cases of l 2 3 and find that the critical behaviour remains unchanged they strongly suggest that this is true for any l rigorous evidence is given to support this conjecture

on the scaling of the chemical distance in long range percolation models we consider the unoriented long range percolation on z d in dimensions d 1 where distinct sites x y z d get connected with probability p xy 0 1 assuming p xy x y s o 1 as x y where s 0 and is a norm distance on z d and supposing that the resulting random graph contains an infinite connected component c we let d x y be the graph distance between x and y measured on c our main result is that for s d 2d d x y log x y o 1 x y c x y where 1 is the binary logarithm of 2d s and o 1 is a quantity tending to zero in probability as x y besides its interest for general percolation theory this result sheds some light on a question that has recently surfaced in the context of small world phenomena as part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box

diffusion effusion and chaotic scattering an exactly solvable liouvillian dynamics we study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area preserving mapping of an infinitely extended strip onto itself this exactly solvable mapping sustains chaotic behaviors and diffusion processes the relationship between the diffusion coefficient the lyapunov exponent and the entropy per unit time is derived the long lived classical resonances of the liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation in this sense there is a quasi isomorphism between the resonance spectrum of the liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation furthermore we show that a fractal repeller is associated to each non equilibrium state in the isolated and finite multibaker chain the nonequilibrium states are all unstable with respect to the equilibrium validating a weak form of the second principle of thermodynamics for the present dynamical system consequences of nonequilibrium fractals on classical measurements are discussed we then describe the open multibaker chain as a scattering system fractal properties of chaotic scattering are here shown to be related to diffusion in the chain

fluctuation theorem for hidden entropy production elimination of seemingly unnecessary variables in markovian models may cause a difference in the value of irreversible entropy production between the original and reduced dynamics we show that such difference which we call the hidden entropy production obeys an integral fluctuation theorem if all variables are time reversal invariant or if the density function is symmetric with respect to the change of sign of the time reversal antisymmetric variables the theorem has wide applicability since the proposed condition is mostly satisfied in the case where the hidden fast variables are equilibrated the main consequence of this theorem is that the entropy production decreases by the coarse graining procedure by contrast in the case where a stochastic process is obtained by coarse graining a deterministic and reversible dynamics the entropy production may increase implying that the integral fluctuation theorem should not hold for such reductions we reveal with an explicit example that the nonequilibrated time reversal antisymmetric variables play a crucial role in distinguishing these two cases thus guaranteeing the consistency of the presented theorem

time reversibility and nonequilibrium thermodynamics of second order stochastic processes nonequilibrium thermodynamics of a general second order stochastic system is investigated we prove that at steady state under inversion of velocities the condition of time reversibility over the phase space is equivalent to the antisymmetry of spatial flux and the symmetry of velocity flux then we show that the condition of time reversibility alone cannot always guarantee the maxwell boltzmann distribution comparing the two conditions together we find that the frictional force naturally emerges as the unique odd term of the total force at thermodynamic equilibrium and is followed by the einstein relation the two conditions respectively correspond to two previously reported different entropy production rates in the case where the external force is only position dependent the two entropy production rates become one we prove that such an entropy production rate can be decomposed into two non negative terms expressed respectively by the conditional mean and variance of the thermodynamic force associated with the irreversible velocity flux at any given spatial coordinate in the small inertia limit the former term becomes the entropy production rate of the corresponding overdamped dynamics while the anomalous entropy production rate originates from the latter term furthermore regarding the connection between the first law and second law we find that in the steady state of such a limit the anomalous entropy production rate is also the leading order of the boltzmann factor weighted difference between the spatial heat dissipation densities of the underdamped and overdamped dynamics while their unweighted difference always tends to vanish

one dimensional percolation problems with further neighbour bonds transfer matrix approach the transfer matrix method is used to find the exact expressions for the correlation length in the critical region for both one dimensional site and bond percolation problems with bonds connecting lth nearest neighbours for any finite l for the site percolation the correlation length exponent v is found to be l consistent with the result obtained from the generating function method for the bond percolation we find v l l 1 2

model simplification and loss of irreversibility in this paper we reveal a general relationship between model simplification and irreversibility based on the model of continuous time markov chains with time scale separation according to the topological structure of the fast process we divide the states of the chain into the transient states and the recurrent states we show that a two time scale chain can be simplified to a reduced chain in two different ways removal of the transient states and aggregation of the recurrent states both the two operations will lead to a decrease in the entropy production rate and its adiabatic part and will keep its nonadiabatic part the same this suggests that although model simplification can retain almost all the dynamic information of the chain it will lose some thermodynamic information as a trade off

mean field behavior for long and finite range ising model percolation and self avoiding walk we consider self avoiding walk percolation and the ising model with long and flnite range by means of the lace expansion we prove mean fleld behavior for these models if d 2 fi 2 for self avoiding walk and the ising model and d 3 fi 2 for percolation where d denotes the dimension and fi the power law decay exponent of the coupling function we provide a simplifled analysis of the lace expansion based on the trigonometric approach in borgs et al 13

simple random walk on long range percolation clusters i heat kernel bounds in this paper we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process for any d 1 and for any exponent s in d d 2 wedge 2d giving the rate of decay of the percolation process we show that the return probability decays like t d s d up to logarithmic corrections where t denotes the time the walk is run our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box the bounds and accompanying understanding of the geometry of the cluster play a crucial role in the companion paper crawford and sly in simple randomwalk on long range percolation clusters ii scaling limit 2010 where we establish the scaling limit of the random walk to be stable levy motion

hamiltonian derivation of a detailed fluctuation theorem we analyze the microscopic evolution of a system undergoing a far from equilibrium thermodynamic process explicitly accounting for the degrees of freedom of participating heat reservoirs we derive a hybrid result similar in form to both the fluctuation theorem and a statement of detailed balance we relate this result to the steady state fluctuation theorem and to a free energy relation valid far from equilibrium

mean field renormalisation group calculations for directed percolation on the square lattice a simple extrapolation technique for mean field renormalisation group calculations is described application of this method to large cell calculations for percolation on the directed square lattice yields pc 0 7067 or 0 0001 site problem and pc 0 6454 or 0 0002 bond problem

what is the probability of connecting two points the two terminal reliability known as the pair connectedness or connectivity function in percolation theory may actually be expressed as a product of transfer matrices in which the probability of operation of each link and site is exactly taken into account when link and site probabilities are p and it obeys an asymptotic power law behaviour for which the scaling factor is the transfer matrix s eigenvalue of largest modulus the location of the complex zeros of the two terminal reliability polynomial exhibits structural transitions as 0 1

mean field theory of percolation with application to surface effects a probabilistic formulation of mean field theory for percolation in a general medium is given the theory covers the whole density range and above the critical density provides the necessary generalisation of the random walk approximation application to surface percolation enables the calculation of scaling functions and critical exponents for the ordinary surface special and extraordinary transitions considered by previous authors to the context of ferromagnetism

entropy production in stochastic systems with fast and slow time scales the behavior of stochastic systems with interacting fast and slow degrees of freedom is investigated both for discrete and continuous processes effective equations that govern the process on the slow timescale are derived by asymptotic methods both for the propagator and the entropy production of the systems it is found that in general the result of the limiting procedure for entropy does not coincide with the one defined for the effective slow process and features an additional contribution the specific conditions under which such a correction does or does not arise are stated and the general explicit form of this remnant entropy production is offered finally the fluctuation theorems that are satisfied by this additional term are given

the mixing time of a random walk on a long range percolation cluster in pre sierpinski gasket we consider a random graph created by the long range percolation on the nth stage finite subset of a fractal lattice called the pre sierpinski gasket the long range percolation is a stochastic model in which any pair of two points is connected by a random bond independently on the random graph obtained as above we consider a discrete time random walk we show that the mixing time of the random walk is of order 2 s d n if d s 2d in a sense here s is a parameter which determines the order of probabilities that random bonds exist and d log 3 log 2 is the hausdorff dimension of the pre sierpinski gasket

lattice decorations and one dimensional percolation it is well known that in long range one dimensional percolation the critical exponents depend on the range a family of decorations is discussed which have range dependent exponents in one dimension although the corresponding decorations in higher dimension preserve the values of the exponents these decorations clarify the unusual nature of one dimensional percolation

transfer matrix approach to the one dimensional percolation problem the transfer matrix method is used to find the exact phase diagrams and the correlation length exponents v for the one dimensional site bond and site bond percolation problems with bonds connecting the lth nearest neighbours l up to 3 for the site percolation the authors results agree with the exact result obtained from the generating function method v l while for the bond percolation they found much richer critical phenomena if all the l bonds have equal occupation probability their results predict v l l 1 2