Free energy in a Markovian model of a lattice spin system

A Markov process which may be thought of as a classical lattice spin system is considered. States of the system are probability measures on the configuration space, and we study the evolution of the free energy of these states with time. It is proved that for all initial states the free energy is nonincreasing and that it strictly decreases from any initial state which is shift invariant but not an equilibrium state. Finally we show that the state of the system converges weakly to the set of Gibbsian Distributions for the given interaction, and that all shift invariant equilibrium states are Gibbsian Distributions.This work was done while the author was a postdoctoral fellow in the Adolph C. and Mary Sprague Miller Institute for Basic Research in Science.     

On the definition of phase transition

It is shown that there exists a phase transition associated with a singularity of the free energy for a model such that for all temperatures the equilibrium state is unique and thus stable with respect to boundary perturbations. It is also shown on this model that there exist phase transitions without symmetry breakdown, which can be related to a phase transition with symmetry breakdown on an equivalent model.     

Existence and uniqueness of equilibrium states for some spin and continuum systems

The one to one correspondence between the existence of a unique equilibrium state and the differentiability of the free energy density with respect to the external field previously shown for Ising ferromagnetis is extendend to higher valued spin systems as well as to continuum systems satisfying the Fortuin, Kasteleyn and Ginibre inequalities. In particular this is shown to hold for a mixture ofA –B particles in which there is no interaction between like particles and a repulsion between unlike particles. Where the derivative of the free energy is discontinuous there are at least two equilibrium states.Supported in part by Air Force Grant n. 732430.     

An Inverse Problem in Quantum Statistical Physics

We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density n(x)? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium (canonical ensemble) with a quantum chemical potential.     

积与积减原理

文章分析了重力势能、引力势能、电荷势能、化学势能、热量势能、质量积、动量积等多种势能,发现它们均可表达为一种守恒广延量和对应的强度量的乘积,因此可将其统一定义为"积".基于积这一概念,文章得到了孤立系统内守恒广延量传递过程的积减原理,即孤立系统内进行的守恒广延量传递过程中系统的积总是减小的.进一步,文章还基于积的概念发展了孤立系统和封闭系统的势平衡判据,发现孤立系统达到势平衡状态时,系统的积达到最小值(最小积原理);当封闭系统达到势平衡状态时,系统的准自由积达到最小值(最小准自由积原理).上述结论应用于传热学中即可得到热量传递过程的(火积)减原理及相应的热平衡判据.与热力学中的核心概念熵相对应,由于物理量(火积)可以描述传热过程的不可逆性,作为传热过程的优化准则,度量系统的无序度,并给出系统的热平衡判据,因此(火积)是传热学中的核心概念. 关键词： 势能 积 积减原理 平衡判据     

The probabilistic-informational opacity functional,Jaynes's principle,and distances to equilibrium of an evolving system

A new probabilistic-informational concept, earlier constructed by Mugur-Schächter, is further developed. Associated with Jaynes's principle, this concept permits one to define a measure for the distance between the state of a system evolving under stable constraints and the equilibrium with these constraints. An illustration is given for a gas evolving in a thermostated box. It appears that the free energy of the gas estimates the distance to equilibrium, the estimation being defined in abstract informational-probabilistic terms.     

Reconstructing Free Energy Profiles from Nonequilibrium Relaxation Trajectories

Reconstructing free energy profiles is an important problem in bimolecular reactions, protein folding or allosteric conformational changes. Nonequilibrium trajectories are readily measured experimentally, but their statistical significance and relation to equilibrium system properties still call for rigorous methods of assessment and interpretation. Here we introduce methods to compute the equilibrium free energy profile of a given variable from a set of short nonequilibrium trajectories, obtained by externally driving a system out of equilibrium and subsequently observing its relaxation. This protocol is not suitable for the Jarzynski equality since the irreversible work on the system is instantaneous. Assuming that the variable of interest satisfies an overdamped Langevin equation, which is frequently used for modeling biomolecular processes, we show that the trajectories sample a nonequilibrium stationary distribution that can be calculated in closed form. This allows for the estimation of the free energy via an inversion procedure that is analogous to that used in equilibrium and bypasses more complicated path integral methods, which we derive for comparison. We generalize the inversion procedure to systems with a diffusion constant that depends on the reaction coordinate, as is the case in protein folding, as well as to protocols in which the trajectories are initiated at random points. Using only a statistical pool of tens of synthetic trajectories, we demonstrate the versatility of these methods by reconstructing double and multi-well potentials, as well as a proposed profile for the hydrophobic collapse of a protein.     

On the inverse problem in statistical mechanics

For quantum spin systems it is known that for a suitable space of potentials the equilibrium states areW*-dense in the set of all translation invariant states. The problem discussed in this paper is how to recognize such equilibrium states and how to find the corresponding potential. A necessary and sufficient condition for a state to be an equilibrium state for some potential is given in Sect. 3.     

Statistical mechanics of quantum spin systems. III

In the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous functionP defined on a Banach space of translationally invariant interactions. We prove that each tangent functional to the graph ofP defines a set of translationally invariant thermodynamic expectation values. More precisely each tangent functional defines a translationally invariant state over a suitably chosen algebra of observables, i. e., an equilibrium state. Properties of the set of equilibrium states are analysed and it is shown that they form a dense set in the set of all invariant states over . With suitable restrictions on the interactions, each equilibrium state is invariant under time-translations and satisfies the Kubo-Martin-Schwinger boundary condition. Finally we demonstrate that the mean entropy is invariant under time-translations.     

An Analogue of the Kac—Wakimoto Formula¶and Black Hole Conditional Entropy

A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac and Wakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the Connes—St?rmer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function of the logarithm of an integer once the initial state or the final state is taken fixed. Received: 31 May 1996 / Accepted: 8 November 1996     

Conditional Expectations Relative to a Product State and the Corresponding Standard Potentials

For a lattice system with a finite number of Fermions and spins on each lattice point, conditional expectations relative to an even product state (such as Fermion Fock vacuum) are introduced and the corresponding standard potential for any given dynamics, or more generally for any given time derivative (at time 0) of strictly local operators, is defined, with the case of the tracial state previously treated as a special case. The standard potentials of a given time derivative relative to different product states are necessarily different but they are shown to give the same set of equilibrium states, where one can compare states satisfying the variational principle (for translation invariant states) or the local thermodynamical stability or the Gibbs condition, all in terms of the standard potential relative to different even product states.     

Thermodynamic Cost of Reliability and Low Temperatures: Tightening Landauer's Principle and the Second Law

Landauer's principle states that the erasure of one bit of information requires thefree energy kT ln 2. We argue that the reliability of the bit erasure process isbounded by the accuracy inherent in the statistical state of the energy source(the resources) driving the process. We develop a general framework describingthe thermodynamic worth of the resources with respect to reliable bit erasureor good cooling. This worth turns out to be given by the distinguishability of theresource's state from its equilibrium state in the sense of a statistical inferenceproblem. Accordingly, Kullback—Leibler relative information is a decisivequantity for the worth of the resource's state. Due to the asymmetry of relativeinformation, the reliability of the erasure process is bounded rather by the relativeinformation of the equilibrium state with respect to the actual state than by therelative information of the actual state with respect to the equilibrium state (whichis the free energy up to constants).     

Independent variables in foam clusters

We find, by counting the degrees of freedom of two-dimensional bubble clusters (finite or periodic) of given topology and bubble areas, that the Plateau laws determine a unique configuration of a finite free cluster, but allow an infinite number of configurations of a periodic cluster. Each of these configurations is associated with a particular strain (stress) state of the cluster; there is in general one unstrained configuration, which corresponds to the minimum of the (surface) energy. Configurations of given topology that satisfy Plateau's laws may only exist in certain ranges of bubble area ratios and/or strains. Received 31 May 2001 and Received in final form 12 September 2001     

Uniqueness, stability and Hessian eigenvalues for two-dimensional bubble clusters

A recent conjecture on two-dimensional foams suggested that for fixed topology with given bubble areas there is a unique state of stable equilibrium. We present counter-examples, consisting of a ring of bubbles around a central one, which refute this conjecture. The discussion centres on a novel form of instability which causes symmetric clusters to become distorted. The stability of these bubble clusters is examined in terms of the Hessian of the energy. Received 8 November 2001     

Relative equilibria for the generalized rigid body

This paper gives necessary and sufficient conditions for the (n-dimensional) generalized free rigid body to be in a state of relative equilibrium. The conditions generalize those for the case of the three-dimensional free rigid body, namely that the body is in relative equilibrium if and only if its angular velocity and angular momentum align, that is, if the body rotates about one of its principal axes. For the n-dimensional rigid body in the Manakov formulation, these conditions have a similar interpretation. We use this result to state and prove a generalized Saari’s Conjecture (usually stated for the N-body problem) for the special case of the generalized rigid body.     

On the uniqueness of the Invariant Equilibrium State and surface tension

Symmetric Equilibrium States and their properties under duality transformation are investigated. Necessary and sufficient conditions are derived for equilibrium states to be transformed into equilibrium states by duality. It is shown that ferromagnetic systems satisfying those conditions have correlation functions bounded by those corresponding to the (+) and free boundary conditions. It is then proved than any Invariant Equilibrium State of a ferromagnetic system is transformed into an equilibrium state by duality and is thus unique if the states defined by the (+), and free boundary conditions coincide on the symmetric algebra. The existence of surface tension between two pure phases is established.     

Evaluating Anisotropic Surface Energies Using the Capillarity Vector Reconstruction Method

By measuring the geometry and crystallography of the three interfaces that meet at grain boundary thermal grooves, it is possible to determine the anisotropy of the surface free energy. Previously, the surface energy of MgO at 1400°C in air was approximated by a truncated double Fourier series with coefficients that were determined by fitting the observations to Herring's condition for local equilibrium at a triple junction. The purpose of this paper is to describe an alternative analysis of the same data set that is not limited by an assumed functional form of the surface energy. In this case, the space of surface characters is discretized and each orientation is associated with a capillarity vector (according to the Cahn–Hoffmann definition). The set of capillarity vectors that most closely satisfies the condition for local equilibrium at each triple junction is then determined by an iterative method. The relative surface free energies derived from this analysis are more anisotropic than those derived from the series fit and more consistent with the observed faceting of MgO in air at 1400°C. The relative surface energies of the low index planes are ₁₁₀/₁₀₀ = 1.07±0.04 and ₁₁₁/₁₀₀ = 1.17±0.04.     

A completely solvable model of the nonlinear Boltzmann equation

In one space and one time dimension, a model of the nonlinear Boltzmann equation is presented that is exactly solvable for all initial conditions. Furthermore, this model has the following desirable properties: (i) conservation of the number of particles; (ii) energy conservation; (iii) nonlinearity; (iv) positivity of distribution functions; and (v) unique equilibrium state (for any given density) which is approached as t → ∞ for most physically interesting initial conditions. Some of the simple properties of this model are studied.