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American Journal of Physics

Approach to Equilibrium

Steve P. Heims

Published Sep 1, 1965• 18 citations
Publisher

Abstract

The time-dependent behavior of an isolated system consisting of many spins in a solid lattice is studied. An interaction of the form Bjkσjxσkx is assumed between any pair (j, k) of spins. The system is treated without making any approximations; it is one of the few many-body systems whose time-evolution can be treated exactly. The behavior is fully determined by the initial density matrix and the distribution of values Bjk. By considering a variety of distributions for the Bjk, and a variety of initial density matrices, one finds cases (initial random phasing and particular continuous spectra) where the system relaxes monotonically to a metastable equilibrium; in other cases (coherent initial state or nearest-neighbor coupling only) the system oscillates indefinitely without relaxation; in still other cases (initial random phasing and N → ∞) the system relaxes in an oscillatory way to the metastable equilibrium.

DOI
10.1119/1.1972193

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The time-dependent behavior of an isolated system consisting of many spins in a solid lattice is studied. An interaction of the form Bjkσjxσkx is assumed between any pair (j, k) of spins. The system is treated without making any approximations; it is one of the few many-body systems whose time-evolution can be treated exactly. The behavior is fully determined by the initial density matrix and the distribution of values Bjk. By considering a variety of distributions for the Bjk, and a variety of initial density matrices, one finds cases (initial random phasing and particular continuous spectra) where the system relaxes monotonically to a metastable equilibrium; in other cases (coherent initial state or nearest-neighbor coupling only) the system oscillates indefinitely without relaxation; in still other cases (initial random phasing and N → ∞) the system relaxes in an oscillatory way to the metastable equilibrium.
A detailed investigation is made of the approach to equilibrium of system of spins with I=12I=\frac{1}{2}I=21​. The spin-spin interaction ${\ensuremath{\Sigma}}_{k}{\ensuremath{\Sigma}}_{j<k}{B}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}$ with Bjk{B}_{\mathrm{jk}}Bjk​ constant is treated exactly, while the additional interaction ${\ensuremath{\Sigma}}_{k}{\ensuremath{\Sigma}}_{j<k}{L}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}$ with Ljk{L}_{\mathrm{jk}}Ljk​ depending on lattice vibrations is treated by means of the assumption of sufficiently short correlation times for the Ljk{L}_{\mathrm{jk}}Ljk​ operators. All correlations between the Ljk{L}_{\mathrm{jk}}Ljk​ are included. The effect of the correlations usually neglected is expressed in terms of a sum over states somewhat resembling an Ising-model partition function with the time replacing interaction strength. The oscillatory relaxation via the Bjk{B}_{\mathrm{jk}}Bjk​ and the monotonic relaxation via the phonons compete with each other; interference effects between the two relaxation modes also occur; the origin and nature of the irreversibility are very different for the two relaxation modes.
We study numerically and analytically isolated interacting quantum systems that are taken out of equilibrium instantaneously (quenched). The probability of finding the initial state in time, the so-called fidelity, decays fastest for systems described by full random matrices, where simultaneous many-body interactions are implied. In the realm of realistic systems with two-body interactions, the dynamics is slower and depends on the interplay between the initial state and the Hamiltonian characterizing the system. The fastest fidelity decay in this case is Gaussian and can persist until saturation. A simple general picture, in which the fidelity plays a central role, is also achieved for the short-time dynamics of few-body observables. It holds for initial states that are eigenstates of the observables. We also discuss the need to reassess analytical expressions that were previously proposed to describe the evolution of the Shannon entropy. Our analyses are mainly developed for initial states that can be prepared in experiments with cold atoms in optical lattices.
A general analytical theory of temporal relaxation processes in isolated quantum systems with many degrees of freedom is elaborated, which unifies and substantially amends several previous approximations. Specifically, the Fourier transform of the initial energy distribution is found to play a key role, which is furthermore equivalent to the so-called survival probability in case of a pure initial state. The main prerequisite is the absence of any notable transport currents, caused for instance by some initially unbalanced local densities of particles, energy, and so on. In particular, such a transportless relaxation scenario naturally arises when both the system Hamiltonian and the initial non-equilibrium state do not exhibit any spatial inhomogeneities on macroscopic scales. A further requirement is that the relaxation must not be notably influenced by any approximate (but not exact) constant of motion or metastable state. The theoretical predictions are compared with various experimental and numerical results from the literature.
We provide an overview of our numerical and analytical studies of isolated interacting quantum systems that are abruptly quenched out of equilibrium instantaneously. We describe the relaxation process to a new equilibrium and obtain lower bounds for the relaxation time of full random matrices and realistic systems with two-body interactions. We show that the size of the time fluctuations after relaxation decays exponentially with system size for systems without too many degeneracies. We also discuss the conditions for thermalization and demonstrate that it can happen after local and global quenches in space. The analyses are developed for systems, initial states, and few-body observables accessible to experiments with optical lattices.
A general analytical theory of temporal relaxation processes in isolated quantum systems with many degrees of freedom is elaborated, which unifies and substantially amends several previous approximations. Specifically, the Fourier transform of the initial energy distribution is found to play a key role, which is furthermore equivalent to the so-called survival probability in case of a pure initial state. The main prerequisite is the absence of any notable transport currents, caused for instance by some initially unbalanced local densities of particles, energy, and so on. In particular, such a transportless relaxation scenario naturally arises when both the system Hamiltonian and the initial non-equilibrium state do not exhibit any spatial inhomogeneities on macroscopic scales. A further requirement is that the relaxation must not be notably influenced by any approximate (but not exact) constant of motion or metastable state. The theoretical predictions are compared with various experimental and numerical results from the literature.
Impurities coupled to superconductors offer a controlled platform to understand the interplay between superconductivity, many-body interactions, and nonequilibrium physics. In the equilibrium situation, local interactions at the impurity induce a transition from the spin-singlet to the spin-doublet ground state, resulting in a supercurrent sign reversal ($0\text{\ensuremath{-}}\ensuremath{\pi}$ transition). In this work, we apply the exact time-dependent density matrix renormalization group method to simulate the transient dynamics of such superconducting systems. We also use a perturbative approximation to analyze their properties at longer times. These two methods agree for a wide range of parameters. In a phase-biased situation, the system gets trapped in a metastable state characterized by a lower supercurrent compared to the equilibrium case. We show that local Coulomb interactions do not provide an effective relaxation mechanism for the initially trapped quasiparticles. In contrast, other relaxation mechanisms, such as coupling to a third normal lead, make the impurity spin relax for parameter values corresponding to the equilibrium 0 phase. For parameters corresponding to the equilibrium $\ensuremath{\pi}$ phase the impurity converges to a spin-polarized stationary state. Similar qualitative behavior is found for a voltage-biased junction, which provides an effective relaxation mechanism for the trapped quasiparticles in the junction.