Entropy Production Of Systems Out Of Equilibrium And Statistical Property Of Eletrolytes

One major discovery of non-equilibrium statistical mechanics in recent few decades is that entropy production can be defined at the level of individual dynamic trajectory of slow variables.For several types of non-equilibrium processes,entropy production along a dynamic path has been identified as logarithmic ratio of the probability of the path to that of its Time-reversal.It appears that all non-equilibrium work relations and fluctuation theorems can be derived from this entropy production formula in a back-of-envelop.More importantly,this formula may play a pivotal role in a dynamic ensemble theory of non-equilibrium statistical mechanics.In this work we synthesize various earlier relevant works and show that the validity of this formula relies only on two conditions:(1)Unitarity of microscopic dynamics,and(2)Initial state as a Boltzmann macrostate.We demonstrate that this formula is applicable to systems either classical and quantum,either weakly or strongly coupled to environment,either autonomous or driven by time-dependent parameters,with or without separation of time-scales.When applied to non-equilibrium steady states,however,care must be taken.Unless the process is Markovian,and all slow variables have definite Time-reversal symmetry,the condition(2)may not be satisfied,and application of the entropy production formula may leads to error.We give counter-examples where the formula breaks down.Finally we give the covariant formulation of non-linear Langevin equation.For electrolyte,we generalize the Debye charging method to linear response theory for asymmetric electrolyte.We obtain the effective charge distribution for constituent ions of the electrolyte.From this distribution we obtain all the linear response property of the electrolyte.We find that when the density of the ions is high,the screening length and dielectric constant are substantially renormalized due to the ion correlation effect.Furthermore,the valence of larger ions is also renormalized while that of smaller ions almost remain the same.When the density is high enough,charge oscillation appears.The threshold of oscillation is much lower for asymmetric electrolytes than symmetric one.Our result fits well with large-scale Monte Carlo simulations.Finally we study potential in the charged plate problem in asymetric eletrolytes using self-consistent field theory(SCFT).We obtain the nonlinear Poisson-Boltzmann theory to the zeroth order in our perturbation.For asymmetric electrolyte,first order correction leads to secular term.This indicates the break down of regular perturbation method.We remove the secular term using the renormalization group method and obtain a globally well-behave first order approximation.We can subtract the renormalized Debye length and renormalized surface charge density from our result.When the counterions are multivalent and the bare surface charge density is sufficiently large,the renormalized surface charge density may be opposite to the bare one,that is,charge inversion happens.Our results agrees with large MC simulation even when the density of electrolytes is relatively high.