Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeneralized Boltzmann distributions for systems strongly coupled to large finite bath -- a microcanonical approach
149
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The theory of probability shows that, as the fraction $X_n/Yto 0$, the conditional probability for $X_n$, given $X_n+Y in h_{delta}:=[h, h+delta]$, has a limit law $f_{X_n}(x)e^{-psi_n(h_delta)x}$, where $psi_n(h_delta) $ equals to $[partial ln P(Y in y_delta)/partial y]_{y=h}$ plus an additional term, contributed from the correlation between $X_n$ and bath $Y$. By applying this limit law to an isolated composite system consisting of two strongly coupled parts, a system of interest and a large but finite bath, we derive the generalized Boltzmann distribution law for the system of interest in the exponential form of a redefined Hamiltonian and corrected Boltzmann temperature that reflects the modification due to strong system-bath coupling and the large but finite bath.
rate research
Read More
We propose a new concept for the dynamics of a quantum bath, the Chebyshev space, and a new method based on this concept, the Chebyshev space method. The Chebyshev space is an abstract vector space that exactly represents the fermionic or bosonic bath degrees of freedom, without a discretization of the bath density of states. Relying on Chebyshev expansions the Chebyshev space representation of a bath has very favorable properties with respect to extremely precise and efficient calculations of groundstate properties, static and dynamical correlations, and time-evolution for a great variety of quantum systems. The aim of the present work is to introduce the Chebyshev space in detail and to demonstrate the capabilities of the Chebyshev space method. Although the central idea is derived in full generality the focus is on model systems coupled to fermionic baths. In particular we address quantum impurity problems, such as an impurity in a host or a bosonic impurity with a static barrier, and the motion of a wave packet on a chain coupled to leads. For the bosonic impurity, the phase transition from a delocalized electron to a localized polaron in arbitrary dimension is detected. For the wave packet on a chain, we show how the Chebyshev space method implements different boundary conditions, including transparent boundary conditions replacing infinite leads. Furthermore the self-consistent solution of the Holstein model in infinite dimension is calculated. With the examples we demonstrate how highly accurate results for system energies, correlation and spectral functions, and time-dependence of observables are obtained with modest computational effort.
In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature $beta$. The system admits the canonical distribution at inverse temperature $beta$ as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large-system limit, satisfies an effective Boltzmann-type equation.
We develop a Magnus formalism for periodically driven systems which provides an expansion both in the driving term and the inverse driving frequency, applicable to isolated and dissipative systems. We derive explicit formulas for a driving term with a cosine dependence on time, up to fourth order. We apply these to the steady state of a classical parametric oscillator coupled to a thermal bath, which we solve numerically for comparison. Beyond dynamical stabilization at second order, we find that the higher orders further renormalize the oscillator frequency, and additionally create a weakly renormalized effective temperature. The renormalized oscillator frequency is quantitatively accurate almost up to the parametric instability, as we confirm numerically. Additionally, a cut-off dependent term is generated, which indicates the break-down of the hierarchy of time scales of the system, as a precursor to the instability. Finally, we apply this formalism to a parametrically driven chain, as an example for the control of the dispersion of a many-body system.
We consider a kinetic model whose evolution is described by a Boltzmann-like equation for the one-particle phase space distribution $f(x,v,t)$. There are hard-sphere collisions between the particles as well as collisions with randomly fixed scatterers. As a result, this evolution does not conserve momentum but only mass and energy. We prove that the diffusively rescaled $f^varepsilon(x,v,t)=f(varepsilon^{-1}x,v,varepsilon^{-2}t)$, as $varepsilonto 0$ tends to a Maxwellian $M_{rho, 0, T}=frac{rho}{(2pi T)^{3/2}}exp[{-frac{|v|^2}{2T}}]$, where $rho$ and $T$ are solutions of coupled diffusion equations and estimate the error in $L^2_{x,v}$.
The quantitative information on the spectral gaps for the linearized Boltzmann operator is of primary importance on justifying the Boltzmann model and study of relaxation to equilibrium. This work, for the first time, provides numerical evidences on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a collision matrix. The original spectral gap problem is then approximated by a constrained minimization problem, with objective function being the Rayleigh quotient of the collision matrix and with constraints being the conservation laws. A conservation correction then applies. We also showed the convergence of the approximate Rayleigh quotient to the real spectral gap for the case of integrable angular cross-sections. Some distributed eigen-solvers and hybrid OpenMP and MPI parallel computing are implemented. Numerical results on integrable as well as non-integrable angular cross-sections are provided.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
