An equation for the quantum dissipative system in statistical mechanics

A formalism for describing quantum dissipative systems in statistical mechanics is developed. A new equation of the Lindblad type with a quadratic superoperator consisting of Hermitian dissipative operators is derived from the Bloch equation for temperature density matrix using the Feynman integral over the trajectories with a modified Menskii weight functional. By way of example, this equation is solved for a one-dimensional quantum harmonic oscillator with linear dissipation. Applying the projection operator technique, an integral-differential equation for a reduced temperature statistical operator is obtained, which is analogous to the Zwanzig equation in statistical mechanics, and its formal solution is found as a convergent series. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 30–34, December, 2006.     

Issues in the statistical mechanics of steady sedimentation

A critical review is presented of recent experimental and theoretical work on the steady sedimentation of particulate suspensions in viscous fluids. The point of view is that of a practitioner of non-equilibrium statistical physics rather than classical fluid mechanics.     

Solitons in the statistical mechanics of the Toda lattice

Using proper canonical variables of the Toda soliton we develop a formalism to describe a gas of independent solitons. We find that this model gives the exact thermodynamic properties for high temperatures.     

On the equations of state in quantum statistical mechanics

By extending methods previously used to study the equations of state at low temperature, it is shown that the entropy density and the statistical average of a conserved, non-spontaneously violated, charge density can be expanded in terms of integrals over products of many body n-point amplitudes defined for real, continuous frequencies. The general structure of the expansions is described, and it is demonstrated that essentially the same spectral function determines the entropy density and the average charge densities. Certain classes of terms are worked out in detail, and the formal sum of one such class is shown to provide the contributions to the equations of state arising from composite quasiparticles associated with the poles of the n-point amplitudes. [Another term, discussed in many previous works, involves the logarithms of the elementary propagators and yields the contributions to the equations of state coming from elementary quasiparticles.]The Appendices include an extensive study of the analytic properties of many body amplitudes in the frequencies of the external and internal lines. Specialized to zero temperature, these considerations apply to the Feynman diagrams for elementary particle amplitudes.     

氧化锌发光中心的统计力学模型

根据ZnO结构的两种空隙(四面体和多面体),提出了一种发光模型.用统计系综求得了上述两种填隙锌的数量计算公式.     

A non-extensive statistical mechanical approach to define the equilibrium value function in the kinetics of voltage-gated ion channels

We present a model for the steady-state (or equilibrium) behaviors of the voltage-gated ion channels in cell membranes using the non-extensive or generalized statistical mechanics. The equilibrium value function in the gating kinetics of batrachotoxin-modified sodium channels from a squid optic nerve in planar bilayers are calculated for different values of entropic index (q) which characterizes the degree of non-extensivity of Tsallis’ entropy and the fractal structure of the channels. It is found that in the limit q→1, the results of calculation reduce to the results described by the well-known Boltzmann statistics or the extensive physics. For the non-extensive case (q≠1), a small deviation with respect to the Boltzmann curve which was observed in a great variety of physical systems occurred.     

A variational framework for the inverse Henderson problem of statistical mechanics

Tensor network states (TNS) are a powerful approach for the study of strongly correlated quantum matter. The curse of dimensionality is addressed by parametrizing the many-body state in terms of a network of partially contracted tensors. These tensors form a substantially reduced set of effective degrees of freedom. In practical algorithms, functionals like energy expectation values or overlaps are optimized over certain sets of TNS. Concerning algorithmic stability, it is important whether the considered sets are closed because, otherwise, the algorithms may approach a boundary point that is outside the TNS set and tensor elements diverge. We discuss the closedness and geometries of TNS sets, and we propose regularizations for optimization problems on non-closed TNS sets. We show that sets of matrix product states (MPS) with open boundary conditions, tree tensor network states, and the multiscale entanglement renormalization ansatz are always closed, whereas sets of translation-invariant MPS with periodic boundary conditions (PBC), heterogeneous MPS with PBC, and projected entangled pair states are generally not closed. The latter is done using explicit examples like the W state, states that we call two-domain states, and fine-grained versions thereof.

     

Dynamic properties of the Monte Carlo method in statistical mechanics

By means of the Monte Carlo sampling technique the equilibrium thermodynamics of fluids and magnets can be calculated numerically. We show that the questions of convergence and accuracy of this method can be understood in terms of the dynamics of the appropriate stochastic model. Also, we discuss to what extent various choices of transition probabilities lead to different dynamic properties of the system. As examples of applications, we consider Ising and Heisenberg spin systems. The numerical results about the dynamic correlation functions are compared to simple approximations taken from the theory of the kinetic Ising model.     

Wave function of the harmonic oscillator in classical statistical mechanics

The notion of wave function of the classical harmonic oscillator is discussed. The evolution equation for this wave function is obtained using the classical Liouville equation for the probability-distribution function of the harmonic oscillator. The tomographic-probability distribution of the classical oscillator is studied. Examples of the ground-like state and the coherent state of the classical harmonic oscillator are considered.     

Embeddings ofU(1)-current algebras in non-commutative algebras of classical statistical mechanics

Motivated by multiplicativeK-homology, and understanding critical phenomena in some classical statistical mechanical models, we construct actions ofGL() on the operator algebras of V. Jones and Ocneanu, and analyse these in terms of embeddings ofU(1)-current algebras.Partially supported by the Science and Engineering Research Council     

A new approach for the justification of ensembles in quantum statistical mechanics — I

In this and the following paper, a new approach for the justification of ensembles in statistical mechanics is given. The essential physical idea is that a measurement is an average of values arising from disjoint regions in three-space. This idea is given a mathematical basis in terms of a class of operators called local operators, and the first paper is devoted primarily to the development of the properties of local operators. In particular, a complete characterization of the bounded local operators on ₂ spaces of finite measure is given. Two results of importance for statistical mechanics are also derived. First, it is shown that the observables of quantum mechanics are local operators. Second, it is shown that the expectation value of an observable for a pure state can be written formally as an ensemble average. In the following paper, these results are used to develop a new approach for the justification of statistical ensembles.This work was supported in part by research grants from the National Science Foundation and the U.S. Public Health Service. The material of this paper is contained in a doctoral dissertation submitted by the author to the University of Oregon (1969).     

A new approach for the justification of ensembles in quantum statistical mechanics — II

The results of the first paper in this series are generalized to include spin, permutation symmetry, and time dependence. In particular, the question of time invariance of localness in the Heisenberg picture is discussed and it is conjectured that an operator that is initially local will remain local over time. In order to treat macroscopic systems, it is shown that the ensemble decomposition of the previous paper can be used to coarsegrain configuration space. Finally, a physical interpretation of the ensemble decomposition in terms of redundant macroscopic information is used to give a derivation of the generalized microcanonical average.This work was supported in part by research grants from the National Science Foundation and the U.S. Public Health Service. Some of the material in this paper is contained in a doctoral dissertation submitted by the author to the University of Oregon (1969).     

Unusual constraints in the quantum statistical mechanics of Josephson junction systems

In order to apply quantum statistical mechanics to systems composed of Josephson junctions, the unconventional constraint of fixed macroscopic wave function magnitudes on either side of a junction must be accommodated. In order to use this information, the density matrix formalism must be extended to deal directly with probability distributions over general quantum states. As a result, in thermal equilibrium, the explicit temperature dependence becomes modified from the trivial 1/kT factors.     

Correlation functions and the uniqueness of the state in classical statistical mechanics

A general criterion is derived which assures the uniqueness of the state of a classical statistical mechanical system in terms of a given system of correlation functions. The criterion is \(\sum\limits_k {(m_{k + j}^A )} ^{ - 1/k} = \infty\) for allj and all bounded setsA, where $$m_k^A = (k!)^{ - 1} \int\limits_A \cdots \int\limits_A {\varrho _k } (x_1 , \ldots ,x_k )dx_1 \ldots dx_1 .$$