# Stochastic equations for thermodynamics

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 J. Chem. Soc., Faraday Trans. 93 (1997) [arxiv ] Stochastic equations for thermodynamics Roumen Tsekov Department of Physical Chemistry, University of Sofia, 1164 Sofia, ulgaria The applicability of stochastic differential equations to thermodynamics is considered and a new form, different from the classical Itô and Stratonovich forms, is introduced. It is shown that the new presentation is more appropriate for the description of thermodynamic fluctuations. The range of validity of the oltzmann-einstein principle is also discussed and a generalised alternative is proposed. oth expressions coincide in the small fluctuation limit, providing a normal distribution density. The controversy concerning the proper meaning and the appropriate application in physics of the stochastic differential equations is full of pitfalls. One is the unclear separation between the macroscopic value of a physical quantity and its fluctuations. The aim of the present paper is to clarify some problems that arise in the application of stochastic differential equations in thermodynamics. It is well-known [1] that many processes in nature can be described by the following stochastic equation dx A( X ) dt ( X ) dw ( t) named after Paul Langevin. Here X is the quantity under observation, t is time, A and are deterministic functions and W is a random Wiener process. The integral form of the Langevin equation is t t X ( t) X (0) A( X ) ds ( X ) dw (1) 0 0 where the notation indicates a peculiar definition of the integrals over a Wiener process. As was mentioned, the main problem in eq. (1) is the interpretation of the last integral. Following standard mathematics, it can be expressed by a Riemann sum as n n 1 n lim [ X ( t)][ W( t) W( t)] N N N N 1 N n 0

2 where is a real number between 0 and 1. In contrast to the usual integrals, which are independent of the value of, this expression depends substantially on the choice of the middle point in the time intervals. This is evident from the corresponding Fokker-Planck equation describing the evolution of the probability density P( x, t ) () (1 ) tp x[ AP ( / ) x( P)] which can be rigorously derived from eq. (1) [1]. In the literature, there are two world-wide accepted choices for leading to two different forms of eq. (): the Itô form with = 0 [] and the Stratonovich form with = 1/ [3]. From the mathematical point of view, both forms are correct but their application to physics and chemistry generates some problems [4]. The main difficulty is related to the right attribution of physical meaning to the functions A and [5]. The latter are not independent and their relationship, being the subject of the fluctuation-dissipation relation [6], is given by the equilibrium solution Px ( ) of eq. () (1 ) x xln A P As seen, the -problem disappears if is constant. For this reason, let us start with a pure random process obeying the following Langevin equation dx A( X ) dt dw (3) Regardless the value of, the corresponding Fokker-Planck equation is P ( AP P / ) and the drift term A is related to the equilibrium probability density as follows t x x A ln P (4) x Let us now introduce a new random variable being a deterministic function of X. The equilibrium probability density Px ( ) is connected to Py ( ) via the relation [7] P( x) P( y) C( y) (5) where C( ) dx. Multiplying eq. (3) by C and using eqs. (4) and (5) one obtains a new stochastic differential equation for the random evolution of driven by a multiplicative white noise ( / )( ln ) d C d PC dt C dw (6)

3 Equation (6) is a particular sample of the Langevin equation with C. Its corresponding Fokker-Planck equation is (1 ) tp y[ C Pd y ln PC C y( C P)]/ which is correct in the equilibrium state ( P P) only if = 1/. Hence, (7) tp y( C P y ln P / P) / and the correct form of eq. (6) is the Stratonovich one ( / )( ln ) 1/ d C d PC dt C dw (8) The fact that the Stratonovich form corresponds to the usual rules of mathematics is proven by the Wong-Zakai theorem [8]. However, there are many alternative forms of eq. (6) which provide the same Fokker-Planck equation. Two examples are d C d PC dt C dw (9) ( / )( ln ) 0 d C d P dt C dw (10) ( / )( ln ) 1 Equation (9) is the Itô form of the Langevin equation. Its advantage is that the equilibrium average value of the drift term is zero. For this reason it is suitable for juxtaposition with the corresponding macroscopic equations [7]. Equation (10) is a new one. Its convenience is related to the proportionality of the drift term to the gradient of the equilibrium distribution density. In this sense, it could be appropriate for non-equilibrium thermodynamics. All equations (8-10) are exact. There is no physics there and using any one of them is a matter of convenience. In general, the transition to another, without change in the corresponding Fokker-Planck equation, is ruled by ( dw ) ( d / dx ) dt [1]. According to thermodynamics, the characteristic function of a closed system at constant temperature T is the free energy F. Any spontaneous process in the system leads to decrease of F to its minimal value corresponding to the equilibrium state. Hence, if Z is a variable parameter of the system, the rate of free energy decrease can be presented as d F ( F) d Z 0 (11) t Z t

4 On the other hand, according to the non-equilibrium thermodynamics, there is a linear relationship between the rate change of Z and the gradient of the free energy d Z M ( Z) F (1) t Z 1 where M is the resistance of the system. Owing to the positive definition of M inequality (11) is always fulfilled. Equation (1) is not stochastic and describes only the irreversible evolution toward equilibrium without accounting for the fluctuations. The latter are not subject of the second law of thermodynamics. The difference between the equilibrium free energy and the conditional free energy for a given value of Z is proportional to the logarithm of the equilibrium probability density to observe this fluctuation value [6] F( Z) F( Z) k T ln P( Z) (13) Introducing this expression in eq. (1) one can easily obtain that the differential stochastic equation for the Z -fluctuations should be written in the form dz k TM ( d ln P) dt k TM 1 dw (14) Z Equation (14) is a particular sample of eq. (10) and the corresponding Fokker-Planck equation is P ( k TMP ln P / P) t z z As is seen, the most appropriate value of in the description of thermodynamic fluctuations is 1 and this is not surprising. In physics, the casualty principle is very important. The only value of which accounts for the whole influence of the Wiener process to the contemporary evolution of the variable under observation is 1. Other -values correspond to a physically unacceptable influence of fluctuations taking place after the current time to the moment state of the considered process. A common mistake here is to treat eq. (1) as the average product of eq. (14). This is only true if M is constant. Equation (1) is not exact. It is a result of the Second Law and does not take into account the thermodynamic fluctuations which lead to the free energy increase. The exact equation is (14) which reduces to eq. (1) in the absence of fluctuations. Finally, we shell pay attention to a problem of range of validity of the oltzmann-einstein principle (13). It is obvious that the oltzmann-einstein principle is not general because it is not invariant against non-linear change of the variable by the law (5). The only case for which it is always satisfied is the small fluctuation limit corresponding to a Gaussian distribution function.

5 Hereafter, an alternative point of view is presented which seems to be more general. According to the statistical mechanics, the equilibrium probability density for fluctuations of Z( ) can be calculated by the canonical Gibbs distribution F H P( z) ( z Z)exp( ) d kt where H( ) is Hamilton function of the system and the free energy F is given by exp( F H ) exp( ) d k T k T (15) A more appropriate statistical quantity for description of the fluctuations is the characteristic function which is defined as the Fourier image of the probability density F H isktz ( s) exp( isz) P( z) dz exp( ) exp( ) d k T k T (16) According to the thermodynamics, the average value Z of the fluctuating thermodynamic parameter is equal to the first derivative of the free energy F with respect to its thermodynamically conjugated quantity, i.e. Z F. For instance, in simple systems could be temperature, volume or number of particles and the corresponding Z quantities are entropy, pressure or chemical potentials. From this point of view and eq. (15), the characteristic function (16) can be expressed as F( ) F( iskt ) ( s) exp[ ] kt and an alternative to the oltzmann-einstein principle (13) is F( ) F( isk T) k Tln ( s) which is invariant to any change of the variable. It is clear that for Gaussian fluctuations both the expressions coincide. Owing to the minimal value of the free energy at equilibrium, the necessary condition 0 1 follows from the equation above. The logarithm of the characteristic function is the so-called cumulant generating function [6] using which a number of correlation characteristics of the system fluctuations can be calculated. The results obtained in the present paper possess rather methodological character rather than being a concrete new knowledge for Nature. The foregoing discussion aims to acquaint the readers with the common difficulty arising in the use of stochastic differential equations and to

6 prevent them of undesirable mistakes. Since the theory of the equilibrium state is much more developed, the basic checkpoint for any kinetic theory is the reproduction of the equilibrium theory results. The reader can note that this is a good tool for discrimination between the different interpretations of the stochastic equations. Of course, the latter can be obtained from first principles. However, owing to the complexity of the systems this is possible only in very simple cases, e.g. harmonic oscillators. In a previous paper [9] starting from the classical mechanics, we have demonstrated that the dynamics of a rownian particle in solids obeys the following equation dr M ( R)( U) dt k TM ( R) 1 dw R where R is the particle coordinate, UR ( ) is potential and M( R ) is position dependent mobility. As is seen, this equation is a particular example of eq. (14). Any other choice of will reflect in physically undue dependence of the equilibrium distribution on the particle friction. Appendix Nowadays, the choice 1 is known as the Hänggi-Klimontovich form [10]. The goal of the present appendix is to apply the above theory to another interesting example, which is related to dissipative nonlinearity. At large velocities the linear friction regime is usually violated [11] and the friction coefficient of a particle becomes velocity-dependent. A very generic model follows from the theory of activated transport, where the velocity of particle migration under the action of a force f is given by v / c exp( f / k T) / exp( f / k T) / sinh( f / k T) (17) Here c is the average thermal velocity and is the typical length scale of the environment structure or the mean free path in gases. Reversing eq. (17) yields an expression for the friction force f mc v c ( / )arcsinh( / ) (18) It is interesting that the friction coefficient f / mv decreases with increase of the particle velocity. At slow motion v c it is nearly constant c /, while at large velocities the friction force is logarithmically dependent on v, i.e. the Coulomb-Amontons law of solid friction almost holds. In the middle range of velocities the friction force is well approximated by the cubic friction model with ( c v / 6 c) /. Equation (17) is an example of a nonlinear flow-force relationship from non-equilibrium thermodynamics. Accordingly, the stochastic differential equation of a free rownian particle with nonlinear friction coefficient acquires the form

7 dv ( V) Vdt ( V)dW (19) As is seen, the thermal noise in eq. (19) is multiplicative and the corresponding Fokker-Planck equation reads (0) (1 ) tp v[ ( v) vp ( / ) v( P)] The equilibrium solution of the Fokker-Planck equation should be the Maxwell distribution. Substituting P( v) exp( v / c ) / c in eq. (0) provides the fluctuation-dissipation relation (1 ) c c ( v) (1) v Thus, the diffusion coefficient in the velocity space depends essentially on the choice of the middle point. In the Itô case 0 the integration of eq. (1) yields 1 v c v v c dv P v vpdv () 0 exp( / ) ( )exp( / ) ( ) v v Therefore, the Itô diffusion coefficient equals to the friction force averaged for larger velocities over the Maxwell distribution. On the contrary, the Hänggi-Klimontovich diffusion coefficient is purely local, since eq. (1) reduces to at 1. Consequently, the 1 c ( v) kt ( v) / m choice of the middle point is physically evident, which discriminates the stochastic models. The effect of nonlinear friction on the quantum rownian motion was recently described [1]. 1. C.W. Gardiner, Handbook of Stochastic Methods, Springer, erlin, K. Itô, Nagoya Math. J. 3 (1951) R.L. Stratonovich, SIAM J. Control 4 (1966) N.G. van Kampen, J. Stat. Phys. 4 (1981) R. Tsekov, Ann. Univ. Sofia, Fac. Chem. 88 (1995) R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics, Nauka, Moscow, D. Ryter, Z. Phys. 30 (1978) 19; P. Mazur and D. edeaux, Langmuir 8 (199) E. Wong and M. Zakai, Ann. Math. Stat. 36 (1965) R. Tsekov and E. Ruckenstein, J. Chem. Phys. 100 (1994) I.M. Sokolov, Chem. Phys. 375 (010) u.l. Klimontovich, Phys. Usp. 37 (1994) R. Tsekov, Ann. Univ. Sofia, Fac. Phys. 105 (01) 14

### Melting of Ultrathin Lubricant Film Due to Dissipative Heating of Friction Surfaces

ISSN 6-78, Technical Physics, 7, Vol. 5, No. 9, pp. 9. Pleiades Publishing, Ltd., 7. Original Russian Text.V. Khomenko, I.. Lyashenko, 7, published in Zhurnal Tekhnicheskoœ Fiziki, 7, Vol. 77, No. 9, pp.

### Brownian Motion and Langevin Equations

1 Brownian Motion and Langevin Equations 1.1 Langevin Equation and the Fluctuation- Dissipation Theorem The theory of Brownian motion is perhaps the simplest approximate way to treat the dynamics of nonequilibrium

### Lecture 6: Irreversible Processes

Materials Science & Metallurgy Master of Philosophy, Materials Modelling, Course MP4, Thermodynamics and Phase Diagrams, H. K. D. H. Bhadeshia Lecture 6: Irreversible Processes Thermodynamics generally

### Emergent Fluctuation Theorem for Pure Quantum States

Emergent Fluctuation Theorem for Pure Quantum States Takahiro Sagawa Department of Applied Physics, The University of Tokyo 16 June 2016, YITP, Kyoto YKIS2016: Quantum Matter, Spacetime and Information

### Handbook of Stochastic Methods

C. W. Gardiner Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences Third Edition With 30 Figures Springer Contents 1. A Historical Introduction 1 1.1 Motivation I 1.2 Some Historical

### This is a Gaussian probability centered around m = 0 (the most probable and mean position is the origin) and the mean square displacement m 2 = n,or

Physics 7b: Statistical Mechanics Brownian Motion Brownian motion is the motion of a particle due to the buffeting by the molecules in a gas or liquid. The particle must be small enough that the effects

### ONSAGER S RECIPROCAL RELATIONS AND SOME BASIC LAWS

Journal of Computational and Applied Mechanics, Vol. 5., No. 1., (2004), pp. 157 163 ONSAGER S RECIPROCAL RELATIONS AND SOME BASIC LAWS József Verhás Department of Chemical Physics, Budapest University

### Brownian Motion: Fokker-Planck Equation

Chapter 7 Brownian Motion: Fokker-Planck Equation The Fokker-Planck equation is the equation governing the time evolution of the probability density of the Brownian particla. It is a second order differential

### Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences

PHYSICAL REVIEW E VOLUME 60, NUMBER 3 SEPTEMBER 1999 Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences Gavin E. Crooks* Department of Chemistry, University

### 8 Lecture 8: Thermodynamics: Principles

8. LECTURE 8: THERMODYNMICS: PRINCIPLES 69 8 Lecture 8: Thermodynamics: Principles Summary Phenomenological approach is a respectable way of understanding the world, especially when we cannot expect microscopic

### Fokker-Planck Equation with Detailed Balance

Appendix E Fokker-Planck Equation with Detailed Balance A stochastic process is simply a function of two variables, one is the time, the other is a stochastic variable X, defined by specifying: a: the

### HIGH FRICTION LIMIT OF THE KRAMERS EQUATION : THE MULTIPLE TIME SCALE APPROACH. Lydéric Bocquet

HIGH FRICTION LIMIT OF THE KRAMERS EQUATION : THE MULTIPLE TIME SCALE APPROACH Lydéric Bocquet arxiv:cond-mat/9605186v1 30 May 1996 Laboratoire de Physique, Ecole Normale Supérieure de Lyon (URA CNRS 1325),

### Statistical Mechanics of Active Matter

Statistical Mechanics of Active Matter Umberto Marini Bettolo Marconi University of Camerino, Italy Naples, 24 May,2017 Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017

### Stochastic Behavior of Dissipative Hamiltonian Systems with Limit Cycles

Stochastic Behavior of Dissipative Hamiltonian Systems with Limit Cycles Wolfgang Mathis Florian Richter Richard Mathis Institut für Theoretische Elektrotechnik, Gottfried Wilhelm Leibniz Universität Hannover,

### Energy Barriers and Rates - Transition State Theory for Physicists

Energy Barriers and Rates - Transition State Theory for Physicists Daniel C. Elton October 12, 2013 Useful relations 1 cal = 4.184 J 1 kcal mole 1 = 0.0434 ev per particle 1 kj mole 1 = 0.0104 ev per particle

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble The object of this endeavor is to impose a simple probability

### Optimum protocol for fast-switching free-energy calculations

PHYSICAL REVIEW E 8, 7 Optimum protocol for fast-switching free-energy calculations Philipp Geiger and Christoph Dellago* Faculty of Physics, University of Vienna, Boltzmanngasse 5, 9 Vienna, Austria Received

### Thermodynamics of nuclei in thermal contact

Thermodynamics of nuclei in thermal contact Karl-Heinz Schmidt, Beatriz Jurado CENBG, CNRS/IN2P3, Chemin du Solarium B.P. 120, 33175 Gradignan, France Abstract: The behaviour of a di-nuclear system in

### Onsager theory: overview

Onsager theory: overview Pearu Peterson December 18, 2006 1 Introduction Our aim is to study matter that consists of large number of molecules. A complete mechanical description of such a system is practically

### 1 Introduction. 2 Diffusion equation and central limit theorem. The content of these notes is also covered by chapter 3 section B of [1].

1 Introduction The content of these notes is also covered by chapter 3 section B of [1]. Diffusion equation and central limit theorem Consider a sequence {ξ i } i=1 i.i.d. ξ i = d ξ with ξ : Ω { Dx, 0,

### Major Concepts Langevin Equation

Major Concepts Langevin Equation Model for a tagged subsystem in a solvent Harmonic bath with temperature, T Friction & Correlated forces (FDR) Markovian/Ohmic vs. Memory Chemical Kinetics Master equation

### 14. Energy transport.

Phys780: Plasma Physics Lecture 14. Energy transport. 1 14. Energy transport. Chapman-Enskog theory. ([8], p.51-75) We derive macroscopic properties of plasma by calculating moments of the kinetic equation

### Smoluchowski Diffusion Equation

Chapter 4 Smoluchowski Diffusion Equation Contents 4. Derivation of the Smoluchoswki Diffusion Equation for Potential Fields 64 4.2 One-DimensionalDiffusoninaLinearPotential... 67 4.2. Diffusion in an

### VIII.B Equilibrium Dynamics of a Field

VIII.B Equilibrium Dynamics of a Field The next step is to generalize the Langevin formalism to a collection of degrees of freedom, most conveniently described by a continuous field. Let us consider the

### Stochastic Integration and Stochastic Differential Equations: a gentle introduction

Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27 Intro: why Stochastic? Brownian Motion/ Wiener process

### Tomoi Koide (IF,UFRJ)

International Conference of Physics (2016) Tomoi Koide (IF,UFRJ) Pedagogical introduction, Koide et. al., J. Phys. Conf. 626, 012055 (`15) Variational Principle in Class. Mech. b a 2 t F m dx() t I( x)

### Lectures on basic plasma physics: Kinetic approach

Lectures on basic plasma physics: Kinetic approach Department of applied physics, Aalto University April 30, 2014 Motivation Layout 1 Motivation 2 Boltzmann equation (a nasty bastard) 3 Vlasov equation

### Anomalous Transport and Fluctuation Relations: From Theory to Biology

Anomalous Transport and Fluctuation Relations: From Theory to Biology Aleksei V. Chechkin 1, Peter Dieterich 2, Rainer Klages 3 1 Institute for Theoretical Physics, Kharkov, Ukraine 2 Institute for Physiology,

### Numerical Integration of SDEs: A Short Tutorial

Numerical Integration of SDEs: A Short Tutorial Thomas Schaffter January 19, 010 1 Introduction 1.1 Itô and Stratonovich SDEs 1-dimensional stochastic differentiable equation (SDE) is given by [6, 7] dx

### arxiv:cond-mat/ v1 [cond-mat.stat-mech] 1 Aug 2001

Financial Market Dynamics arxiv:cond-mat/0108017v1 [cond-mat.stat-mech] 1 Aug 2001 Fredrick Michael and M.D. Johnson Department of Physics, University of Central Florida, Orlando, FL 32816-2385 (May 23,

### Dynamic model and phase transitions for liquid helium

JOURNAL OF MATHEMATICAL PHYSICS 49, 073304 2008 Dynamic model and phase transitions for liquid helium Tian Ma 1 and Shouhong Wang 2,a 1 Department of Mathematics, Sichuan University, Chengdu, 610064, People

### 15.3 The Langevin theory of the Brownian motion

5.3 The Langevin theory of the Brownian motion 593 Now, by virtue of the distribution (2), we obtain r(t) =; r 2 (t) = n(r,t)4πr 4 dr = 6Dt t, (22) N in complete agreement with our earlier results, namely

### Symmetry of the Dielectric Tensor

Symmetry of the Dielectric Tensor Curtis R. Menyuk June 11, 2010 In this note, I derive the symmetry of the dielectric tensor in two ways. The derivations are taken from Landau and Lifshitz s Statistical

### Linear Response and Onsager Reciprocal Relations

Linear Response and Onsager Reciprocal Relations Amir Bar January 1, 013 Based on Kittel, Elementary statistical physics, chapters 33-34; Kubo,Toda and Hashitsume, Statistical Physics II, chapter 1; and

### 12. MHD Approximation.

Phys780: Plasma Physics Lecture 12. MHD approximation. 1 12. MHD Approximation. ([3], p. 169-183) The kinetic equation for the distribution function f( v, r, t) provides the most complete and universal

### Elementary Lectures in Statistical Mechanics

George DJ. Phillies Elementary Lectures in Statistical Mechanics With 51 Illustrations Springer Contents Preface References v vii I Fundamentals: Separable Classical Systems 1 Lecture 1. Introduction 3

### CONTENTS 1. In this course we will cover more foundational topics such as: These topics may be taught as an independent study sometime next year.

CONTENTS 1 0.1 Introduction 0.1.1 Prerequisites Knowledge of di erential equations is required. Some knowledge of probabilities, linear algebra, classical and quantum mechanics is a plus. 0.1.2 Units We

### Stochastic Calculus in Physics

Journal of Statistical Physics, Voi. 46, Nos. 5/6, 1987 Stochastic Calculus in Physics Ronald F. Fox 1 Received October 30, 1986 The relationship of the Ito-Stratonovich stochastic calculus to studies

### Lecture 8. The Second Law of Thermodynamics; Energy Exchange

Lecture 8 The Second Law of Thermodynamics; Energy Exchange The second law of thermodynamics Statistics of energy exchange General definition of temperature Why heat flows from hot to cold Reading for

### Some Topics in Stochastic Partial Differential Equations

Some Topics in Stochastic Partial Differential Equations November 26, 2015 L Héritage de Kiyosi Itô en perspective Franco-Japonaise, Ambassade de France au Japon Plan of talk 1 Itô s SPDE 2 TDGL equation

### are divided into sets such that the states belonging

Gibbs' Canonical in Ensemble Statistical By Takuzo Distribution Mechanics. Law SAKAI (Read Feb. 17,1940) 1. librium The distribution state is obtained law for an assembly various methods. of molecules

### arxiv: v2 [cond-mat.stat-mech] 16 Mar 2012

arxiv:119.658v2 cond-mat.stat-mech] 16 Mar 212 Fluctuation theorems in presence of information gain and feedback Sourabh Lahiri 1, Shubhashis Rana 2 and A. M. Jayannavar 3 Institute of Physics, Bhubaneswar

### Nonlinear Dynamics and Fluctuations of Dissipative Toda Chains 1

Journal of Statistical Physics, Vol. 101, Nos. 12, 2000 Nonlinear Dynamics and Fluctuations of Dissipative Toda Chains 1 W. Ebeling, 2 U. Erdmann, 2, 3 J. Dunkel, 2 and M. Jenssen 2, 4 Received November

### 2m + U( q i), (IV.26) i=1

I.D The Ideal Gas As discussed in chapter II, micro-states of a gas of N particles correspond to points { p i, q i }, in the 6N-dimensional phase space. Ignoring the potential energy of interactions, the

### CHAPTER 3 LECTURE NOTES 3.1. The Carnot Cycle Consider the following reversible cyclic process involving one mole of an ideal gas:

CHATER 3 LECTURE NOTES 3.1. The Carnot Cycle Consider the following reversible cyclic process involving one mole of an ideal gas: Fig. 3. (a) Isothermal expansion from ( 1, 1,T h ) to (,,T h ), (b) Adiabatic

### Lecture 8. The Second Law of Thermodynamics; Energy Exchange

Lecture 8 The Second Law of Thermodynamics; Energy Exchange The second law of thermodynamics Statistics of energy exchange General definition of temperature Why heat flows from hot to cold Reading for

### arxiv: v1 [cond-mat.stat-mech] 3 Apr 2007

A General Nonlinear Fokker-Planck Equation and its Associated Entropy Veit Schwämmle, Evaldo M. F. Curado, Fernando D. Nobre arxiv:0704.0465v1 [cond-mat.stat-mech] 3 Apr 2007 Centro Brasileiro de Pesquisas

### Modelling across different time and length scales

18/1/007 Course MP5 Lecture 1 18/1/007 Modelling across different time and length scales An introduction to multiscale modelling and mesoscale methods Dr James Elliott 0.1 Introduction 6 lectures by JAE,

### The First Principle Calculation of Green Kubo Formula with the Two-Time Ensemble Technique

Commun. Theor. Phys. (Beijing, China 35 (2 pp. 42 46 c International Academic Publishers Vol. 35, No. 4, April 5, 2 The First Principle Calculation of Green Kubo Formula with the Two-Time Ensemble Technique

### Relativistic quantum Brownian motion

Annuaire de l Universite de Sofia St. Kliment Ohridski, Faculte de Physique v. 105, 2012 Relativistic quantum Brownian motion Roumen Tsekov Department of Physical Chemistry, Faculty of Chemistry, St. Kliment

### Inverse Langevin approach to time-series data analysis

Inverse Langevin approach to time-series data analysis Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Universidade de Brasília Saratoga Springs, MaxEnt 2007 Outline 1 Motivation 2 Outline 1 Motivation

### Stochastic Thermodynamics of Langevin systems under time-delayed feedback control

Japan-France Joint Seminar (-4 August 25) New Frontiers in Non-equilibrium Physics of Glassy Materials Stochastic Thermodynamics of Langevin systems under time-delayed feedback control M.L. Rosinberg in

### Introduction to Statistical Thermodynamics

Cryocourse 2011 Chichilianne Introduction to Statistical Thermodynamics Henri GODFRIN CNRS Institut Néel Grenoble http://neel.cnrs.fr/ Josiah Willard Gibbs worked on statistical mechanics, laying a foundation

### Thermal and Statistical Physics Department Exam Last updated November 4, L π

Thermal and Statistical Physics Department Exam Last updated November 4, 013 1. a. Define the chemical potential µ. Show that two systems are in diffusive equilibrium if µ 1 =µ. You may start with F =

### Eli Pollak and Alexander M. Berezhkovskii 1. Weizmann Institute of Science, Rehovot, Israel. Zeev Schuss

ACTIVATED RATE PROCESSES: A RELATION BETWEEN HAMILTONIAN AND STOCHASTIC THEORIES by Eli Pollak and Alexander M. Berehkovskii 1 Chemical Physics Department Weimann Institute of Science, 76100 Rehovot, Israel

### Lecture 6: Bayesian Inference in SDE Models

Lecture 6: Bayesian Inference in SDE Models Bayesian Filtering and Smoothing Point of View Simo Särkkä Aalto University Simo Särkkä (Aalto) Lecture 6: Bayesian Inference in SDEs 1 / 45 Contents 1 SDEs

### Introduction to a few basic concepts in thermoelectricity

Introduction to a few basic concepts in thermoelectricity Giuliano Benenti Center for Nonlinear and Complex Systems Univ. Insubria, Como, Italy 1 Irreversible thermodynamic Irreversible thermodynamics

### Modified Maxwell-Garnett equation for the effective transport coefficients in inhomogeneous media

J. Phys. A: Math. Gen. 3 (998) 7227 7234. Printed in the UK PII: S0305-4470(98)93976-2 Modified Maxwell-Garnett equation for the effective transport coefficients in inhomogeneous media Juris Robert Kalnin

### Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism

Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger

### Space Plasma Physics Thomas Wiegelmann, 2012

Space Plasma Physics Thomas Wiegelmann, 2012 1. Basic Plasma Physics concepts 2. Overview about solar system plasmas Plasma Models 3. Single particle motion, Test particle model 4. Statistic description

### Thermodynamics for small devices: From fluctuation relations to stochastic efficiencies. Massimiliano Esposito

Thermodynamics for small devices: From fluctuation relations to stochastic efficiencies Massimiliano Esposito Beijing, August 15, 2016 Introduction Thermodynamics in the 19th century: Thermodynamics in

### Preface. Preface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1 Introduction 1

xi Contents Preface Preface to the Third Edition Preface to the Second Edition Preface to the First Edition v vii viii ix 1 Introduction 1 I GENERAL THEORY OF OPEN QUANTUM SYSTEMS 5 Diverse limited approaches:

### Nonequilibrium thermodynamics at the microscale

Nonequilibrium thermodynamics at the microscale Christopher Jarzynski Department of Chemistry and Biochemistry and Institute for Physical Science and Technology ~1 m ~20 nm Work and free energy: a macroscopic

### New Physical Principle for Monte-Carlo simulations

EJTP 6, No. 21 (2009) 9 20 Electronic Journal of Theoretical Physics New Physical Principle for Monte-Carlo simulations Michail Zak Jet Propulsion Laboratory California Institute of Technology, Advance

### arxiv:cond-mat/ v2 [cond-mat.stat-mech] 23 Feb 1998

Multiplicative processes and power laws arxiv:cond-mat/978231v2 [cond-mat.stat-mech] 23 Feb 1998 Didier Sornette 1,2 1 Laboratoire de Physique de la Matière Condensée, CNRS UMR6632 Université des Sciences,

### The effect of the spatial correlation length in Langevin. micromagnetic simulations

F043, version 1, 30 May 2001 The effect of the spatial correlation length in Langevin micromagnetic simulations V. Tsiantos a, W. Scholz a, D. Suess a, T. Schrefl a, J. Fidler a a Institute of Applied

### Thermodynamics: A Brief Introduction. Thermodynamics: A Brief Introduction

Brief review or introduction to Classical Thermodynamics Hopefully you remember this equation from chemistry. The Gibbs Free Energy (G) as related to enthalpy (H) and entropy (S) and temperature (T). Δ

### How the Nonextensivity Parameter Affects Energy Fluctuations. (Received 10 October 2013, Accepted 7 February 2014)

Regular Article PHYSICAL CHEMISTRY RESEARCH Published by the Iranian Chemical Society www.physchemres.org info@physchemres.org Phys. Chem. Res., Vol. 2, No. 2, 37-45, December 204. How the Nonextensivity

### Ana María Cetto, Luis de la Peña and Andrea Valdés Hernández

Ana María Cetto, Luis de la Peña and Andrea Valdés Hernández Instituto de Física, UNAM EmQM13, Vienna, 3-6 October 2013 1 Planck's law as a consequence of the zero-point field Equilibrium radiation field

### 2 Equations of Motion

2 Equations of Motion system. In this section, we will derive the six full equations of motion in a non-rotating, Cartesian coordinate 2.1 Six equations of motion (non-rotating, Cartesian coordinates)

### MESOSCOPIC QUANTUM OPTICS

MESOSCOPIC QUANTUM OPTICS by Yoshihisa Yamamoto Ata Imamoglu A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Toronto Singapore Preface xi 1 Basic Concepts

### Table of contents Lecture 1: Discrete-valued random variables. Common probability distributions and their properties. Bernoulli trials. Binomial, dist

NPTEL COURSE PHYSICAL APPLICATIONS OF STOCHASTIC PROCESSES V. Balakrishnan Department of Physics, Indian Institute of Technology Madras Chennai 600 036, India This course comprises 9 lectures. The notes

### Efficient numerical integrators for stochastic models

Efficient numerical integrators for stochastic models G. De Fabritiis a, M. Serrano b P. Español b P.V.Coveney a a Centre for Computational Science, Department of Chemistry, University College London,

### Entropic forces in Brownian motion

Entropic forces in Brownian motion Nico Roos* Leiden Observatory, PO Box 9513, NL-2300 RA Leiden, The Netherlands Interest in the concept of entropic forces has risen considerably since E. Verlinde proposed

### Phase Transitions. µ a (P c (T ), T ) µ b (P c (T ), T ), (3) µ a (P, T c (P )) µ b (P, T c (P )). (4)

Phase Transitions A homogeneous equilibrium state of matter is the most natural one, given the fact that the interparticle interactions are translationally invariant. Nevertheless there is no contradiction

### Statistical Properties of a Ring Laser with Injected Signal and Backscattering

Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 87 92 c International Academic Publishers Vol. 35, No. 1, January 15, 2001 Statistical Properties of a Ring Laser with Injected Signal and Backscattering

### Comparison of continuous and discrete-time data-based modeling for hypoelliptic systems

Comparison of continuous and discrete- data-based modeling for hypoelliptic systems Fei Lu, Kevin K. Lin, and Alexandre J. Chorin arxiv:1605.02273v2 [math.na] 7 Dec 2016 Abstract We compare two approaches

### The concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.

The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes

### ROBERT BROWN AND THE POLLEN STUFF

ROBERT BROWN AND THE POLLEN STUFF P. Hänggi Institut für Physik Universität Augsburg Robert Brown (1773-1858) This is the view Brown obtained in 1828, when he first recognised the cell nucleus. It shows

### In this section, thermoelasticity is considered. By definition, the constitutive relations for Gradθ. This general case

Section.. Thermoelasticity In this section, thermoelasticity is considered. By definition, the constitutive relations for F, θ, Gradθ. This general case such a material depend only on the set of field

### The Kelvin-Planck statement of the second law

The Kelvin-Planck statement of the second law It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work Q W E =ΔE net net net, mass

### A critical remark on Planck s model of black body

A critical remark on Planck s model of black body Andrea Carati Luigi Galgani 01/12/2003 Abstract We reexamine the model of matter radiation interaction considered by Planck in his studies on the black

### IV. Classical Statistical Mechanics

IV. Classical Statistical Mechanics IV.A General Definitions Statistical Mechanics is a probabilistic approach to equilibrium macroscopic properties of large numbers of degrees of freedom. As discussed

### THERMODYNAMICS Lecture 5: Second Law of Thermodynamics

HERMODYNAMICS Lecture 5: Second Law of hermodynamics Pierwsza strona Second Law of hermodynamics In the course of discussions on the First Law of hermodynamics we concluded that all kinds of energy are

### Lecture 12: Detailed balance and Eigenfunction methods

Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility),

### Gibb s free energy change with temperature in a single component system

Gibb s free energy change with temperature in a single component system An isolated system always tries to maximize the entropy. That means the system is stable when it has maximum possible entropy. Instead

### List of Comprehensive Exams Topics

List of Comprehensive Exams Topics Mechanics 1. Basic Mechanics Newton s laws and conservation laws, the virial theorem 2. The Lagrangian and Hamiltonian Formalism The Lagrange formalism and the principle

### OPTIMAL PERTURBATION OF UNCERTAIN SYSTEMS

Stochastics and Dynamics, Vol. 2, No. 3 (22 395 42 c World Scientific Publishing Company OPTIMAL PERTURBATION OF UNCERTAIN SYSTEMS Stoch. Dyn. 22.2:395-42. Downloaded from www.worldscientific.com by HARVARD

### 2.6 The Membrane Potential

2.6: The Membrane Potential 51 tracellular potassium, so that the energy stored in the electrochemical gradients can be extracted. Indeed, when this is the case experimentally, ATP is synthesized from

### Einstein s Approach to Planck s Law

Supplement -A Einstein s Approach to Planck s Law In 97 Albert Einstein wrote a remarkable paper in which he used classical statistical mechanics and elements of the old Bohr theory to derive the Planck

### Thermodynamic condition for equilibrium between two phases a and b is G a = G b, so that during an equilibrium phase change, G ab = G a G b = 0.

CHAPTER 5 LECTURE NOTES Phases and Solutions Phase diagrams for two one component systems, CO 2 and H 2 O, are shown below. The main items to note are the following: The lines represent equilibria between

### Quantum Grand Canonical Ensemble

Chapter 16 Quantum Grand Canonical Ensemble How do we proceed quantum mechanically? For fermions the wavefunction is antisymmetric. An N particle basis function can be constructed in terms of single-particle

### Electrical Transport in Nanoscale Systems

Electrical Transport in Nanoscale Systems Description This book provides an in-depth description of transport phenomena relevant to systems of nanoscale dimensions. The different viewpoints and theoretical

### Kolmogorov Equations and Markov Processes

Kolmogorov Equations and Markov Processes May 3, 013 1 Transition measures and functions Consider a stochastic process {X(t)} t 0 whose state space is a product of intervals contained in R n. We define

### arxiv: v1 [q-bio.mn] 6 Nov 2015

Realization of Waddington s Metaphor: Potential Landscape, Quasi-potential, A-type Integral and Beyond Peijie Zhou 1 and Tiejun Li 1 arxiv:1511.288v1 [q-bio.mn] 6 Nov 215 1 LMAM and School of Mathematical

### MARKOVIANITY OF A SUBSET OF COMPONENTS OF A MARKOV PROCESS

MARKOVIANITY OF A SUBSET OF COMPONENTS OF A MARKOV PROCESS P. I. Kitsul Department of Mathematics and Statistics Minnesota State University, Mankato, MN, USA R. S. Liptser Department of Electrical Engeneering

### arxiv:physics/ v2 [physics.class-ph] 18 Dec 2006

Fluctuation theorem for entropy production during effusion of an ideal gas with momentum transfer arxiv:physics/061167v [physicsclass-ph] 18 Dec 006 Kevin Wood 1 C Van den roeck 3 R Kawai 4 and Katja Lindenberg

### From time series to superstatistics

From time series to superstatistics Christian Beck School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E 4NS, United Kingdom Ezechiel G. D. Cohen The Rockefeller University,