Using the Renyi entropy to describe quantum dissipative systems in statistical mechanics

We develop a formalism for describing quantum dissipative systems in statistical mechanics based on the quantum Renyi entropy. We derive the quantum Renyi distribution from the principle of maximum quantum Renyi entropy and differentiate this distribution (the temperature density matrix) with respect to the inverse temperature to obtain the Bloch equation. We then use the Feynman path integral with a modified Mensky functional to obtain a Lindblad-type equation. From this equation using projection operators, we derive the integro-differential equation for the reduced temperature statistical operator, an analogue of the Zwanzig equation in statistical mechanics, and find its formal solution in the form of a series in the class of summable functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 444–453, September, 2008.     

Some properties of entropy of order α and type β

In a recent paper, we defined entropy of order α and type β. For α = 1, it reduces to Renyi’s² entropy of order α, while for α = 1, β= 1 and for a complete probability distribution, it reduces to Shannon’s definition of entropy. In this paper we have studied some properties of this generalised entropy. We have also discussed the variability of Renyi’s entropy in the continuous case with co-ordinate systems and invariance of transinformation of order α and type β for linear transformations.     

Fine-grained and coarse-grained entropy in problems of statistical mechanics

We consider dynamical systems with a phase space Γ that preserve a measure μ. A partition of Γ into parts of finite μ-measure generates the coarse-grained entropy, a functional that is defined on the space of probability measures on Γ and generalizes the usual (ordinary or fine-grained) Gibbs entropy. We study the approximation properties of the coarse-grained entropy under refinement of the partition and also the properties of the coarse-grained entropy as a function of time. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 120–137, April, 2007.     

Dynamics of quantum entropy maximization for finite-level systems

The problem of constructing models for the statistical dynamics of finite-level quantum mechanical systems is considered. The maximum entropy principle formulated by E.T. Jaynes in 1957 and asserting that the entropy of any physical system increases until it attains its maximum value under constraints imposed by other physical laws is applied. In accordance with this principle, the von Neumann entropy is taken for the objective function; a dynamical equation describing the evolution of the density operator in finite-level systems is derived by using the speed gradient principle. In this case, physical constraints are the mass conservation law and the energy conservation law. The stability of the equilibrium points of the system thus obtained is investigated. By using LaSalle’s theorem, it is shown that the density function tends to a Gibbs distribution, under which the entropy attains its maximum. The method is exemplified by analyzing a finite system of identical particles distributed between cells. Results of numerical simulation are presented.     

Maximum entropy method for solving operator equations of the first kind

The maximum entropy method for linear ill-posed problems with modeling error and noisy data is considered and the stability and convergence results are obtained. When the maximum entropy solution satisfies the “source condition”, suitable rates of convergence can be derived. Considering the practical applications, ana posteriori choice for the regularization parameter is presented. As a byproduct, a characterization of the maximum entropy regularized solution is given.     

On the stability of the equilibrium states for Hamiltonian dynamical systems arising in non–equilibrium thermodynamics

In this paper we consider a thermodynamic system with an internal state variable, and study the stability of its equilibrium states by exploiting the reduced entropy inequality. Remarkably, we derive a Hamiltonian dynamical system ruling the evolution of the system in a suitable thermodynamic phase space. The use of the Hamiltonian formalism allows us to prove the equivalence of the asymptotic stability at constant temperature, at constant entropy and at constant energy, thus extending some classical results by Coleman and Gurtin (J. Chem. Phys., 47, 597–613, 1967).     

Computation of the ε entropy of the space of continuous functions with the Hausdorff metric

The number K_p,q, i.e., the number of (p, q) corridors of closed domains which are convex in the vertical direction, consist of elementary squares of the integral lattice, are situated within a rectangle of the size q × p, and completely cover the side of length p of this rectangle under projection is computed. The asymptotic (K_p,q/q²)^1/p → λ, as p, q → ∞, where λ = 0.3644255… is the maximum root of the equation₁F₁(-1/2 − 1/(16λ), 1/2, 1/(4λ)) = 0,₁F₁ being the confluence hypergeometric function, is established. These results allow us to compute the ε entropy of the space of continuous functions with the Hausdorff metric. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 39–50, January, 1977.     

Stochastic model of phase transition and metastability

The evolution of a system with phase transition is simulated by a Markov process whose transition probabilities depend on a parameter. The change of the stationary distribution of the Markov process with a change of this parameter is interpreted as a phase transition of the system from one thermodynamic equilibrium state to another. Calculations and computer experiments are performed for condensation of a vapor. The sample paths of the corresponding Markov process have parts where the radius of condensed drops is approximately constant. These parts are interpreted as metastable states. Two metastable states occur, initial (gaseous steam) and intermediate (fog). The probability distributions of the drop radii in the metastable states are estimated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 94–106, April, 2000.     

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup P_t. A fundamental and still largely open problem is the understanding of the long time behavior of δ_ηP_t when the initial configuration η is sampled from a highly disordered state ν (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular b-ary tree , we study the above problem for the Ising and hard core gas (independent sets) models on . If ν is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove ν-almost sure weak convergence of δ_ηP_t to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time t. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.     

Coarse-grained entropy in dynamical systems

Let M be the phase space of a physical system. Consider the dynamics, determined by the invertible map T: M → M, preserving the measure µ on M. Let ν be another measure on M, dν = ρdµ. Gibbs introduced the quantity s(ρ) = ?∝ρ log ρdµ as an analog of the thermodynamical entropy. We consider a modification of the Gibbs (fine-grained) entropy the so called coarse-grained entropy. First we obtain a formula for the difference between the coarse-grained and Gibbs entropy. The main term of the difference is expressed by a functional usually referenced to as the Fisher information. Then we consider the behavior of the coarse-grained entropy as a function of time. The dynamics transforms ν in the following way: ν → ν _n , dν _n = ρ ○ T ^?n dµ. Hence, we obtain the sequence of densities ρ _n = ρ ○ T ^?n and the corresponding values of the Gibbs and the coarse-grained entropy. We show that while the Gibbs entropy remains constant, the coarse-grained entropy has a tendency to a growth and this growth is determined by dynamical properties of the map T. Finally, we give numerical calculation of the coarse-grained entropy as a function of time for systems with various dynamical properties: integrable, chaotic and with mixed dynamics and compare these calculation with theoretical statements.     

Physical entropy, information entropy and their evolution equations

Inspired by the evolution equation of nonequilibrium statistical physics entropy and the concise statistical formula of the entropy production rate, we develop a theory of the dynamic information entropy and build a nonlinear evolution equation of the information entropy density changing in time and state variable space. Its mathematical form and physical meaning are similar to the evolution equation of the physical entropy: The time rate of change of information entropy density originates together from drift, diffusion and production. The concise statistical formula of information entropy production rate is similar to that of physical entropy also. Furthermore, we study the similarity and difference between physical entropy and information entropy and the possible unification of the two statistical entropies, and discuss the relationship among the principle of entropy increase, the principle of equilibrium maximum entropy and the principle of maximum information entropy as well as the connection between them and the entropy evolution equation.     

Strong entropy for system of isentropic gas dynamics

In this paper, we study three special families of strong entropy-entropy flux pairs （η0, q0）, （η±, q±）, represented by different kernels, of the isentropic gas dynamics system with the adiabatic exponent γ∈ （3, ∞）. Through the perturbation technique through the perturbation technique, we proved, we proved the H^-1 compactness of ηit ＋ qix, i = 1, 2, 3 with respect to the perturbation solutions given by the Cauchy problem （6） and （7）, where （ηi, qi） are suitable linear combinations of （η0, q0）, （η±, q±）.     

On directional entropy functions

Given aZ ²-process, the measure theoretic directional entropy function,h( % MathType!End!2!1!), is defined on % MathType!End!2!1!. We relate the directional entropy of aZ ²-process to itsR ² suspension. We find a sufficient condition for the continuity of directional entropy function. In particular, this shows that the directional entropy is continuous for aZ ²-action generated by a cellular automaton; this finally answers a question of Milnor [Mil]. We show that the unit vectors whose directional entropy is zero form aG _δ subset ofS ¹. We study examples to investigate some properties of directional entropy functions. This research is supported in part by BSRI and KOSEF 95-0701-03-3.     

Generalized entropy of the Heisenberg spin chain

We consider the XY quantum spin chain in a transverse magnetic field. We consider the Rényi entropy of a block of neighboring spins at zero temperature on an infinite lattice. The Rényi entropy is essentially the trace of some power α of the density matrix of the block. We calculate the entropy of the large block in terms of Klein’s elliptic λ-function. We study the limit entropy as a function of its parameter α. We show that the Rényi entropy is essentially an automorphic function with respect to a certain subgroup of the modular group. Using this, we derive the transformation properties of the Rényi entropy under the map α → α ⁻¹ .     

Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians

On the adjacency algebra of a distance-regular graph we introduce an analogue of the Gibbs state depending on a parameter related to temperature of the graph. We discuss a scaling limit of the spectral distribution of the Laplacian on the graph with respect to the Gibbs state in the manner of central limit theorem in algebraic probability, where the volume of the graph goes to ∞ while the temperature tends to 0. In the model we discuss here (the Laplacian on the Johnson graph), the resulting limit distributions form a one parameter family beginning with an exponential distribution (which corresponds to the case of the vacuum state) and consisting of its deformations by a Bessel function. Received: 7 July 1999 / Revised version: 23 February 2000 / Published online: 5 September 2000     

On the thermodynamic foundations of the theory of local-gradient mechanicothermodiffusion systems

In the context of the continuous-thermodynamic approach we generalize the Gibbs equation and obtain the initial relations of local-gradient mechanicothermodiffusion. We state the relation between the thermodynamic flows and forces in the form of functionals. We find influence functions that cause expansion of the phase space that determines the thermodynamic potentials by the gradients of the intensive parameters of the equilibrium state of the system. It is shown that such influence functions are connected with the undamped memory of the body of the action at the initial time. Translated fromMatematychni Metody ta Fizyko-Mekhanichni, Polya, Vol. 41, No. 1, 1998, pp. 62–72.     

Erdős measures,sofic measures,and Markov chains

We consider the random variable ζ = ξ₁ρ+ξ₂ρ²+…, where ξ₁, ξ₂, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξ_i = 0) = p₀, P(ξ_i = 1) = p₁, 0 < p₀ < 1. Let β = 1/ρ be the golden number. The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η₁ρ + η₂ρ² + … where the random variables η_k are {0, 1}-valued and η_kη_k+1 = 0. The infinite random word η = η₁η₂ … η_n … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous. We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure. We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable when ξ₁, ξ₂, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47.     

Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields

We show that the entropy functional exhibits a quasi-factorization property with respect to a pair of weakly dependent σ-algebras. As an application we give a simple proof that the Dobrushin and Shlosmans complete analyticity condition, for a Gibbs specification with finite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several different techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way. Received: 7 July 2000 / Revised version: 10 October 2000 / Published online: 5 June 2001     

The stability of the entropy of degree alpha

In this paper, we first prove that the generalized fundamental equation of information depending on a positive real parameter α, is stable in the sense of Hyers and Ulam provided that α≠1, then we apply this result to prove the stability of a system of functional equations that characterizes the entropy of degree alpha or Havrda-Charvát entropy which has recently often been called the Tsallis entropy.