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Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

We study a microscopic Hamiltonian model describing an N-level quantum system S coupled to an infinitely extended thermal reservoir R. Initially, the system S is in an arbitrary state while the reservoir is in thermal equilibrium at temperature T. Assuming that the coupled system S+R is mixing with respect to the joint thermal equilibrium state, we study the Full Counting Statistics (FCS) of the energy transfers S->R and R->S in the process of return to equilibrium. The first FCS describes the increase of the energy of the system S. It is an atomic probability measure, denoted $P_{S,\lambda,t}$, concentrated on the set of energy differences $\sigma(H_S)-\sigma(H_S)$ ($\sigma(H_S)$ is the spectrum of the Hamiltonian of S, $t$ is the length of the time interval during which the measurement of the energy transfer is performed, and $\lambda$ is the strength of the interaction between S and R). The second FCS, $P_{R,\lambda,t}$, describes the decrease of the energy of the reservoir R and is typically a continuous probability measure whose support is the whole real line. We study the large time limit $t\rightarrow\infty$ of these two measures followed by the weak coupling limit $\lambda\rightarrow 0$ and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of $P_{R,\lambda,t}$ by quantum transfer operators.

Energy conservation, counting statistics, and return to equilibrium

We study driven finite quantum systems in contact with a thermal reservoir in the regime in which the system changes slowly in comparison to the equilibration time. The associated isothermal adiabatic theorem allows us to control the full statistics of energy transfers in quasi-static processes. Within this approach, we extend Landauer's Principle on the energetic cost of erasure processes to the level of the full statistics and elucidate the nature of the fluctuations breaking Landauer's bound.

Full statistics of erasure processes: Isothermal adiabatic theory and a statistical Landauer principle

We give results on the absence of singular continuous spectrum of the one-particle Hamiltonian underlying the electronic black box model.

https://www.math.mcgill.ca/jaksic/papers_pdf/EBB-Hamiltonian7.pdf

The Spectral Structure of the Electronic Black Box Hamiltonian

We describe quantum entropic functionals and outline a research program dealing with entropic fluctuations in non-equilibrium quantum statistical mechanics.

Entropic Functionals in Quantum Statistical Mechanics

We discuss the dissipative dynamics of a classical particle coupled to an infinite heat reservoir. We announce a number of results concerning the ergodic properties of this model. The novelty of our approach is that it extends beyond Markovian dynamics to the case where the Langevin equation is driven by colored noise. In this Letter we discuss the dissipative dynamics of a classical particle interacting with a large reservoir. The reservoir is an infinite gas of free classical phonons at thermal equilibrium. We announce a number of results concerning the ergodic properties of the combined system particle + reservoir. In particular, it follows from our analysis that the dynamical system describing this model near thermal equilibrium is strongly mixing. In a recent work ([JP]), we have investigated the problem of thermal relaxation for a finite dimensional Hamiltonian system A coupled to a heat reservoirB. Our assumptions on the systems A and B are quite general and have simple physical interpretations. However the set of hypotheses that we need on the form of the coupling between the two systems is complicated. To avoid many of these technical details, we concentrate here on a simple model. This allows us to give a more transparent exposition of the results presented in [JP]. Using the statistical nature of the reservoir at thermal equilibrium, one can give a reduced probabilistic description of the particle based on a random integro-differential equation: The Langevin equation. The latter departs from the original Newton law by the addition of two terms: A random force describing the direct action of the reservoir on the particle, and a dissipative term arising from the reaction of the reservoir on the motion of the particle. We start by discussing our results in this well-known context (see [FK], [FKM], [LT], [N], [UO] and [W] for additional information on the Langevin equation). Throughout this Letter, the small system A consists of a single particle of unit mass moving on the line under the influence of a confiningC potential V (q) y. The Hamiltonian function of the system A is HA(q p) p2 2 + V (q) with q p R. The equation of motion of the isolated particle is qt pt pt V (qt) (1) Since V is smooth and confining, the initial value problem associated with Equation (1) has a global solution which defines a smooth flow on the phase space R2. The interaction of the particle with the reservoir is described in terms of a friction constant and a coupling function L2(R), a normalized “charge density” (k k 1). Under the influence of the reservoir, which is assumed to be initially in thermal equilibrium at inverse temperature , the equation of motion (1) is modified as follows: qt pt pt V eff (qt) + 2 Z t 0 D(t s)qsds + t (2) y The potential is confining in the sense that lim jqj V (q) and exp( V ) L1(R) for all 0. Ergodic Properties 2 Here t is a stationary Gaussian random process with mean zero and covariance h t si 1C(t s) 1 Z j (k)j2 cos(k(t s)) dk 2 and D(t) C(t) Z kj (k)j2 sin(kt) dk 2 The effective potential is given by Veff (q) V (q)+ 2q2 2, and (k) is the Fourier transform of the function (x). From Equation (2) it is apparent that the reservoir plays a dual role: On one hand, its ability to absorb energy-momentum provides a physical mechanism for dissipation. On the other hand, its thermal fluctuations, encapsulated in t, prevent the particle from relaxing into some stationary state (see [KKS]). Equation (2) is known as the Langevin equation, and the resulting random process as the Ornstein-Uhlenbeck process. The derivation of the Langevin equation from “microscopic” considerations, i.e. from the Hamiltonian formalism describing the combined system A + B, is presented below. The first question, of course, is whether the Langevin equation has a global solution. Our first result is Theorem Suppose that 2 x + x2 s for some s 1. Then for any and for almost all , the initial value problem associated with Equation (2) has a global solution T t (q0 p0) (qt pt) which defines a C1-flow on the phase space R2. Let SA be the set of probability measures on R 2 which are absolutely continuous with respect to the Lebesgue measure. The elements of SA are initial states of the system A. We say that the Ornstein-Uhlenbeck process has the property of return to equilibrium if for each measure SA and for any observable f L (R2) we have lim t Z D f T t E d Z fd A (3) where d A 1 Z A e HA(q p)dqdp Ergodic Properties 3 is the Gibbs canonical ensemble of the system A at the temperature of the reservoir. Clearly, return to equilibrium is equivalent to the mixing property of the Ornstein-Uhlenbeck process with respect to A . Our main result is a set of necessary conditions which ensure that the OU process returns to equilibrium. We note that, except in some special cases [TR], the question of return to equilibrium for the OU process was previously unsolved. The difficulties are related to the presence of memory in (2). A standard approach of these difficulties is based on an ad hoc limiting procedure (see e.g. [FK] or [TH], Example 3.1.9) which leads to the simplified equation qt pt pt V (qt) 2 2 qt + t (4) Here t is the white noise process, h t si 1 (t s) and is the effective friction constant. The resulting OU process is Markovian and standard techniques based on Fokker-Planck equation apply. It is known that the process (4) is mixing with respect to the Gibbs measure A (see [TR]). In [JP] we have proven a result which, specialized to the model (2), translates into the following statement: Theorem Suppose that the conditions of Theorem 1 hold and that j (k)j C (1 + jkj) (5) for some positive constants C and and for all k R. Then, for all non-zero , the Ornstein-Uhlenbeck process (2) has the property of return to equilibrium. We now briefly summarize our program. Our first goal is to construct a differentiable dynamical system (G t ) describing the combined systemA+B near thermal equilibrium. HereG is the phase space, t is the Hamiltonian flow, and is the Gibbs canonical ensemble of the joint system at inverse temperature . Admissible initial states which are “near” thermal equilibrium are probability measures onGwhich are absolutely continuous with respect to . We denote this class of states by S . Observables of the system are elements of the algebra L (G d ). The probabilistic description of the dynamics of the system A is obtained by “integrating out” the variables of the reservoir. If these variables are initially distributed according to the Gibbs canonical ensemble, then this reduced description should yield the Ergodic Properties 4 Langevin equation (2). The problem of return to equilibrium for the combined system can be formalized in the following way. De nition We say that the combined system A + B returns to equilibrium if the dynamical system (G t ) satisfies

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Vojkan Jaksic

Papers

Energy conservation, counting statistics, and return to equilibrium

Full statistics of erasure processes: Isothermal adiabatic theory and a statistical Landauer principle

The Spectral Structure of the Electronic Black Box Hamiltonian

Entropic Functionals in Quantum Statistical Mechanics

Ergodic Properties of the Langevin Equation