A nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution

We discuss a nonlinear model for the relaxation by energy redistribution within an isolated, closed system composed of non-interacting identical particles with energy levels e_i with i=1,2,...,N. The time-dependent occupation probabilities p_i(t) are assumed to obey the nonlinear rate equations tau dp_i/dt=-p_i ln p_i+ alpha(t)p_i-beta(t)e_ip_i where alpha(t) and beta(t) are functionals of the p_i(t)'s that maintain invariant the mean energy E=sum_i e_ip_i(t) and the normalization condition 1=sum_i p_i(t). The entropy S(t)=-k sum_i p_i(t) ln p_i(t) is a non-decreasing function of time until the initially nonzero occupation probabilities reach a Boltzmann-like canonical distribution over the occupied energy eigenstates. Initially zero occupation probabilities, instead, remain zero at all times. The solutions p_i(t) of the rate equations are unique and well-defined for arbitrary initial conditions p_i(0) and for all times. Existence and uniqueness both forward and backward in time allows the reconstruction of the primordial lowest entropy state. The time evolution is at all times along the local direction of steepest entropy ascent or, equivalently, of maximal entropy generation. These rate equations have the same mathematical structure and basic features of the nonlinear dynamical equation proposed in a series of papers ended with G.P.Beretta, Found.Phys., 17, 365 (1987) and recently rediscovered in S. Gheorghiu-Svirschevski, Phys.Rev.A, 63, 022105 and 054102 (2001). Numerical results illustrate the features of the dynamics and the differences with the rate equations recently considered for the same problem in M.Lemanska and Z.Jaeger, Physica D, 170, 72 (2002).Comment: 11 pages, 7 eps figures (psfrag use removed), uses subeqn, minor revisions, accepted for Physical Review

Asymptotics of Quantum Relative Entropy From Representation Theoretical Viewpoint

In this paper it was proved that the quantum relative entropy $D(\sigma \| \rho)$ can be asymptotically attained by Kullback Leibler divergences of probabilities given by a certain sequence of POVMs. The sequence of POVMs depends on $\rho$, but is independent of the choice of $\sigma$.Comment: LaTeX2e. 8 pages. The title was changed from "Asymptotic Attainment for Quantum Relative Entropy

Equilibrium states and their entropy densities in gauge-invariant C*-systems

A gauge-invariant C*-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. In the paper, thermodynamics is studied on such systems and the chemical potential theory developed by Araki, Haag, Kastler and Takesaki is used. As a generalization of quantum spin system, the equivalence of the KMS condition, the Gibbs condition and the variational principle is shown for translation-invariant states. The entropy density of extremal equilibrium states is also investigated in relation to macroscopic uniformity.Comment: 20 pages, revised in March 200

On the comparison of volumes of quantum states

This paper aims to study the \a-volume of \cK, an arbitrary subset of the set of $N\times N$ density matrices. The \a-volume is a generalization of the Hilbert-Schmidt volume and the volume induced by partial trace. We obtain two-side estimates for the \a-volume of \cK in terms of its Hilbert-Schmidt volume. The analogous estimates between the Bures volume and the \a-volume are also established. We employ our results to obtain bounds for the \a-volume of the sets of separable quantum states and of states with positive partial transpose (PPT). Hence, our asymptotic results provide answers for questions listed on page 9 in \cite{K. Zyczkowski1998} for large $N$ in the sense of \a-volume. \vskip 3mm PACS numbers: 02.40.Ft, 03.65.Db, 03.65.Ud, 03.67.M

A set-valued framework for birth-and-growth process

We propose a set-valued framework for the well-posedness of birth-and-growth process. Our birth-and-growth model is rigorously defined as a suitable combination, involving Minkowski sum and Aumann integral, of two very general set-valued processes representing nucleation and growth respectively. The simplicity of the used geometrical approach leads us to avoid problems arising by an analytical definition of the front growth such as boundary regularities. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, it is not local, i.e. for a fixed time instant, growth is the same at each space point

Free energy density for mean field perturbation of states of a one-dimensional spin chain

Motivated by recent developments on large deviations in states of the spin chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expression of free energy density in the presence of a mean field type perturbation. We extend their results from the product state case to the Gibbs state case in the setting of translation-invariant interactions of finite range. In the special case of a locally faithful quantum Markov state, we clarify the relation between two different kinds of free energy densities (or pressure functions).Comment: 29 pages, Section 5 added, to appear in Rev. Math. Phy

A matrix interpolation between classical and free max operations: I. The univariate case

Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings.Comment: 14 page

Large Deviations for a Non-Centered Wishart Matrix

We investigate an additive perturbation of a complex Wishart random matrix and prove that a large deviation principle holds for the spectral measures. The rate function is associated to a vector equilibrium problem coming from logarithmic potential theory, which in our case is a quadratic map involving the logarithmic energies, or Voiculescu's entropies, of two measures in the presence of an external field and an upper constraint. The proof is based on a two type particles Coulomb gas representation for the eigenvalue distribution, which gives a new insight on why such variational problems should describe the limiting spectral distribution. This representation is available because of a Nikishin structure satisfied by the weights of the multiple orthogonal polynomials hidden in the background.Comment: 40 page

Statistical Inference, Distinguishability of Quantum States, And Quantum Entanglement

We argue from the point of view of statistical inference that the quantum relative entropy is a good measure for distinguishing between two quantum states (or two classes of quantum states) described by density matrices. We extend this notion to describe the amount of entanglement between two quantum systems from a statistical point of view. Our measure is independent of the number of entangled systems and their dimensionality.Comment: 11 pages no figures, to appear in Phys. Rev.