Thermodynamics of Small Systems

Evans, Denis J.; Williams, Stephen R.; Searles, Debra J.

doi:10.1002/9783527629374.ch2

Cited by 6 publications

(17 citation statements)

References 106 publications

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“…Whereas if we differentiate Q with respect to T th we see that H E = Q ! T th "Q "T th (21) Since U = H E (internal energy is an entirely mechanical quantity as the First Law shows) and noting that when T = T th = 0 , that A(0) = U(0) = Q(0) , we observe, treating T , T th as integration parameters x, that A and Q satisfy the same differential equations with the same initial x = 0 condition and therefore A(T ) = Q(T th ) and our hypothesis (19) is proved [25,26].…”

Section: (Ie If P[!"g(!(0)t) = A] Is Non-zero For Any a > 0 Then supporting

confidence: 54%

“…That is when T th = T the Helmholtz free energy A(T), at the thermodynamic temperature T, is equal to the value of the statistical mechanical expression Q(T th ) that is defined in (19). From thermodynamics we note that the Helmholtz free energy satisfies the differential equation…”

Section: (Ie If P[!"g(!(0)t) = A] Is Non-zero For Any a > 0 Then mentioning

confidence: 99%

“…An important observation arises from this expression. The necessary and sufficient condition for a distribution function to be time independent is that:• for ergodic systems [20], the dissipation function must be zero everywhere in phase space and,• for nonergodic systems, the dissipation function must be zero everywhere within the ergodic subdomain in which a given sample system is trapped.In reference [15] (also see [19,21]) we proved that we can write averages of arbitrary phase functions asThe derivation of (7), (8) from (4) is called the Dissipation Theorem. This Theorem is extremely general.…”

mentioning

confidence: 99%

“…and the NonEquilibrium Partition Identity [19], i.e. e ,0) implies that the ensemble average is taken over the ensemble defined by the initial distribution f (!, 0) , Eq.…”

mentioning

confidence: 99%

“…The existence of the dissipation function (4) only requires that the initial distribution is normalizable and that ergodic consistency holds. To prove (5) requires two additional conditions: the dynamics must be time reversal symmetric and the initial distribution function must be an even function of the momenta.The TFT leads to a number of corollaries such as the Second Law Inequality [16],and the NonEquilibrium Partition Identity [19], i.e. e ,0) implies that the ensemble average is taken over the ensemble defined by the initial distribution f (!, 0) , Eq.…”

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confidence: 99%

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Dissipation and the relaxation to equilibrium

Evans¹,

Searles²,

Williams³

2009

J. Stat. Mech.

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Using the recently derived Dissipation Theorem and a corollary of the Transient Fluctuation Theorem (TFT), namely the Second Law Inequality, we derive the unique time independent, equilibrium phase space distribution function for an ergodic Hamiltonian system in contact with a remote heat bath. We prove under very general conditions that any deviation from this equilibrium distribution breaks the time independence of the distribution. Provided temporal correlations decay, and the system is ergodic, we show that any nonequilibrium distribution that is an even function of the momenta, eventually relaxes (not necessarily monotonically) to the equilibrium distribution. Finally we prove that the negative logarithm of the microscopic partition function is equal to the thermodynamic Helmholtz free energy divided by the thermodynamic temperature and Boltzmann's constant. Our results complement and extend the findings of modern ergodic theory and show the importance of dissipation in the process of relaxation towards equilibrium. 2The foundations of statistical mechanics are still not completely satisfactory.Textbook derivations of the canonical phase space distribution functions lag a long way behind modern ergodic theory but their derivations fall into two basic categories. The first approach [1][2][3] is to propose a microscopic definition for the entropy and then to show that the standard canonical distribution function can be obtained by maximising the entropy subject to the constraints that the distribution function should be normalized and that the average energy is constant. The choice of the second constraint is completely subjective due to the fact that at equilibrium, the average of any phase function is fixed. This "derivation" is therefore flawed.The second approach begins with Boltzmann's postulate of equal a priori probability in phase space for the microcanonical ensemble [2][3][4][5] and then derives an expression for the most probable distribution of states in a small subsystem within a much larger microcanonical system. A variation on this approach is to simply postulate a microscopic expression for the Helmholtz free energy [3] via the partition function.The relaxation of systems to equilibrium is also fraught with difficulties. The first reasonably general approach to this problem is summarized in the Boltzmann H-theorem.Beginning with the definition of the H-function, Boltzmann proved that the Boltzmann equation for the time evolution of the single particle probability density in an ideal gas, implies a monotonic decrease in the H-function [2,4,6]. There are at least two problems with this. Firstly the Boltzmann equation is only valid for an ideal gas. Secondly and more problematically, unlike Newton's equations the Boltzmann equation itself is not time reversal symmetric. Some recent work on relaxation to equilibrium has been in the context of relating large deviations to relaxation phenomena and some links with fluctuation theorems have been discussed [7]. 3The early 1930's saw significant...

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Section: (Ie If P[!"g(!(0)t) = A] Is Non-zero For Any a > 0 Then supporting

confidence: 54%

Section: (Ie If P[!"g(!(0)t) = A] Is Non-zero For Any a > 0 Then mentioning

confidence: 99%

mentioning

confidence: 99%

“…and the NonEquilibrium Partition Identity [19], i.e. e ,0) implies that the ensemble average is taken over the ensemble defined by the initial distribution f (!, 0) , Eq.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Dissipation and the relaxation to equilibrium

Evans¹,

Searles²,

Williams³

2009

J. Stat. Mech.

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The Evans–Searles Fluctuation Theorem

2016

Fundamentals of Classical Statistical Thermodynamics

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Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

王波

Wang

2012

Appl. Math. Mech.-Engl. Ed.

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The weakly forced vibration of an axially moving viscoelastic beam is investigated. The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved. The nonlinear equations governing the transverse vibration are derived from the dynamical, constitutive, and geometrical relations. The method of multiple scales is used to determine the steady-state response. The modulation equation is derived from the solvability condition of eliminating secular terms. Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation. The stability of nontrivial steady-state response is examined via the Routh-Hurwitz criterion.

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Thermodynamics of Small Systems

Cited by 6 publications

References 106 publications

Dissipation and the relaxation to equilibrium

Dissipation and the relaxation to equilibrium

The Evans–Searles Fluctuation Theorem

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

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