We demonstrate that, for a quantum Markovian semigroup on a finite-dimensional Hilbert space, if it satisfies the standard quantum detailed balance condition, its generator admits a special representation that yields a vanishing entropy production rate. Conversely, if the generator admits a special representation adhering to the condition of thermodynamic consistency and leading to a vanishing entropy production rate, then the corresponding quantum Markovian semigroup must satisfy the standard quantum detailed balance condition. In this context, we adopt the definition of entropy production rate that is motivated by the physics literature and standard for thermodynamically consistent Lindbladians.
We prove that the canonical partition function of a hard-core model with s molecules on a d-dimensional hypercubic lattice with interaction range R is a polynomial in the lattice lengths n_i ( i=1,… ,d ), provided that n_i>Rs for all i. The polynomial has total degree sd, and degree s in each variable. Under periodic boundary conditions, the partition function depends polynomially on the single variable N=∏ _i=1^d n_i . This polynomial characterization provides a unified framework for deriving a variety of recurrence relations satisfied by the partition function.
In this paper, we discuss the asymptotic dynamics of the distribution of the solution process to the non-autonomous McKean–Vlasov stochastic lattice plate equations with a variable time delay defined on the integer set ℤ. First, we rigorously investigate the well-posedness of solutions for these non-autonomous, distribution-dependent stochastic delay lattice plate equations. Subsequently, we establish the existence and uniqueness of pullback measure attractors for this system characterized by the solution operators. Furthermore, as an application of pullback measure attractor, we show that under appropriate conditions, the invariant measure for the system is ergodic and exponentially mixing. Finally, we characterize the upper semi-continuity of pullback measure attractors as the stochastic delay lattice plate system with distribution dependence converges to a system with distribution independence.
This review article discusses wave propagation and heat transport in one-dimensional nonlinear lattices, focusing particularly on the role of symmetry. The concept of continuous translational symmetry in nonlinear lattices has recently been introduced. Furthermore, a nonlinear lattice possessing this symmetry has been explicitly constructed. This paper describes this symmetric lattice and its various properties, focusing especially on the propagation of discrete breather (DB). Effects due to the lattice discreteness emerge in traveling DBs in ordinary nonlinear lattices. In contrast, the constructed lattice removes these discrete effects, showing that the lack of this symmetry is the origin of the discrete effect. The paper also outlines recent progress regarding the construction of nonlinear lattices without umklapp processes, which possess similar symmetry, and their heat transport. Numerical results indicate that this similar symmetry vanishes thermal resistance by eliminating the umklapp process. These findings suggest a unified perspective that symmetry determines both the mobility and the heat transport of nonlinear waves in lattices.
A measure of correlations for statistical systems is advanced, called correlation index. The measure is general, being applicable to quantum and classical, equilibrium and nonequilibrium statistical systems. After formulating the general notion of correlation indices for arbitrary trace-class operators, the calculation of the correlation indices are illustrated by several examples of density operators and reduced density matrices. The computation of the correlation indices for equilibrium systems is exemplified for bosonic systems with the Bose-Einstein condensate, superconducting systems, and for spin systems. Correlation indices for nonequilibrium systems are considered for nonlinear dynamics of spin and pseudospin systems.
