Solvable Many-body Systems Research Articles

Faster Uphill Relaxation in Thermodynamically Equidistant Temperature Quenches.

We uncover an unforeseen asymmetry in relaxation: for a pair of thermodynamically equidistant temperature quenches, one from a lower and the other from a higher temperature, the relaxation at the ambient temperature is faster in the case of the former. We demonstrate this finding on hand of two exactly solvable many-body systems relevant in the context of single-molecule and tracer-particle dynamics. We prove that near stable minima and for all quadratic energy landscapes it is a general phenomenon that also exists in a class of non-Markovian observables probed in single-molecule and particle-tracking experiments. The asymmetry is a general feature of reversible overdamped diffusive systems with smooth single-well potentials and occurs in multiwell landscapes when quenches disturb predominantly intrawell equilibria. Our findings may be relevant for the optimization of stochastic heat engines.

Simulating Quantum Dynamics with Entanglement Mean Field Theory

Exactly solvable many-body systems are few and far between, and the utility of approximate methods cannot be overestimated. Entanglement mean field theory is an approximate method to handle such systems. While mean field theories reduce the many-body system to an effective single-body one, entanglement mean field theory reduces it to a two-body system. And in contrast to mean field theories where the self-consistency equations are in terms of single-site physical parameters, those in entanglement mean field theory are in terms of both single- and two-site parameters. Hitherto, the theory has been applied to predict properties of the static states, like ground and thermal states, of many-body systems. Here we give a method to employ it to predict properties of time-evolved states. The predictions are then compared with known results of paradigmatic spin Hamiltonians.

Generalized Langevin equation revisited: mechanical random force and self-consistent structure

The generalized Langevin equation (GLE) is considered with emphasis put on a mechanical random force , whose time evolution is not natural due to the presence of a projection operator in a propagator. We first derive the GLE for a set of dynamical variables of interest not by formal manipulation of the operator identity as is usually done, but by expressing the random force in terms of natural trajectory in the phase space. Based on this new derivation we observe two structural characteristics, (1) a linear functional interdependence between and and (2) a self-consistent structure (SCS) inherent in the GLE. The SCS gives an iterative procedure to determine and its time correlation function, i.e. a memory function K(t), once we have numerical data for supplied by computer experiments. This method generates and a memory function K(t), nicely for an exactly solvable many-body system after several iterations. Some merits of using the SCS are further discussed in connection with numerical methods to calculate the autocorrelation function of .

Exactly Solvable Many-Body Systems and Pseudo-Hermitian Point Interactions

We study Hamiltonian systems with point interactions and give a systematic description of the corresponding boundary conditions and the spectrum properties for self-adjoint, PT-symmetric systems and systems with real spectra. The integrability of one dimensional many body systems with these kinds of point (contact) interactions are investigated for both bosonic and fermionic statistics.

Hamilton's principal function for the Brownian motion of a particle and the Schrödinger–Langevin equation

The Brownian motion of a particle coupled to a heat bath can be derived from an exactly solvable many-body system. This motion is governed by the Langevin equation in Newtonian mechanics, and by an operator equation formally identical to the Langevin equation when the system composed of the particle plus the heat bath is quantized. To get this formal similarity between the classical and the quantal equations of motion of the particle, the classical dissipative system is formulated in terms of a modified Hamilton–Jacobi equation and is then quantized using the Schrödinger method. The result is identical with the Schrödinger–Langevin equation that has been obtained by quantizing the entire system and then isolating the motion of the particle. The non-linear wave equation describing the motion of a particle subject to conservative and time-dependent forces as well as frictional forces has been applied to the problems of motion of a wave-packet, and of the scattering and trapping of heavy-ions.

Development of a Phase Transition for a Rigorously Solvable Many-Body System

The ferromagnetic transition is considered in detail for the Heisenberg Hamiltonian with all spins coupled equally. For this special model the statistical problem is solved exactly. The transition develops very slowly as the number $N$ of particles in the system is increased; the spin order at the nominal Curie temperature is proportional to ${N}^{\ensuremath{-}\frac{1}{4}}$. The order in the most probable state differs from the mean value of the order, except for $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The connections are studied with the molecular-field, spin-wave, and self-consistent approximations. The failure of the self-consistent approximation is relatively severe as compared with the nearest-neighbor problem. The antiferromagnetic ground state is very close in energy to the N\'eel state.