1,720,977 research outputs found
Spectral gap for an unrestricted Kawasaki type dynamics
Full text linkWe give an accurate asymptotic estimate for the gap of the generator of a particular interacting particle system. The model we consider may be informally described as follows. A certain number of charged particles moves on the segment [1, L] ∩ N according to a Markovian law. If ηk ∈ Zis the charge at a site k ∈ [1,L]∩None unitary charge, positive or negative, jumps to a neighboring site, k ± 1 at a rate which depends on the charge at site k and at site k ± 1. The total charge Lk=1 ηk is preserved by the dynamics, in this sense our dynamics is similar to the Kawasaki dynamics, but in our case there is no restriction on the maximum charge allowed per site. The model is equivalent to an interface dynamics connected with the stochastic Ising model at very low temperature: the “unrestricted solid on solid model”. Thus the results we obtain may be read as results for this model. We give necessary and sufficient conditions to ensure that gap shrinks as L−2, independently of the total charge. We follow the method outlined in some papers by Yau (Lu, Yau (1993), Yau (1994)) where a similar spectral gap is proved for the original Kawasaki dynamics
Spectral Asymptotics for Variational Fractals
No full textA generalization of a classic result of H. Weyl concerning the asymptotics of the spectrum of the Laplace operator is proved for variational fractals. Physically we are studying the density of states for the diffusion through a fractal medium. A variational fractal is a couple (K,E) where K is a self-similar fractal and E is an energy form with some similarity properties connected with those of K. In this class we can find some of the most widely studied families of fractals, such as nested fractals, p.c.f. fractals, the Sierpiński carpet, etc., as well as some regular self-similar Euclidean domains. We find that if r(x) is the number of eigenvalues associated with E smaller than x, then r(x)∼xν/2, where ν is the intrinsic dimension of (K,E)
Equilibrium fluctuations for a one-dimensional interface in the solid on solid approximation.
No full textAn unbounded one-dimensional solid-on-solid model with integer heights is studied. Unbounded here means that there is no a priori restrictions on the discrete gradient of the interface. The interaction Hamiltonian of the interface is given by a finite range part, proportional to the sum of height differences, plus a part of exponentially decaying long range potentials. The evolution of the interface is a reversible Markov process. We prove that if this system is started in the center of a box of size L after a time of order L^3 it reaches, with a very large probability, the top or the bottom of the box
Entropy dissipation estimates in a zero-range dynamics
No full textWe study the exponential decay of relative entropy functionals for zero-range processes on the complete graph. For the standard model with rates increasing at infinity we prove entropy dissipation estimates, uniformly over the number of particles and the number of vertices
Boundary driven Brownian gas
Full text linkWe consider a gas of independent Brownian particles on a bounded in- terval in contact with two particle reservoirs at the endpoints. Due to the Brownian nature of the particles, infinitely many particles enter and leave the system in each time interval. Nonetheless, the dynamics can be constructed as a Markov process with continuous paths on a suitable state space. If λ0 and λ1 are the chemical potentials of the boundary reservoirs, the stationary distribution (reversible if and only if λ0 = λ1) is a Poisson point process with intensity given by the linear inter- polation between λ0 and λ1. We then analyze the empirical flow that it is defined by counting, in a time interval [0,t], the net number of particles crossing a given point x. In the stationary regime we identify its statistics and show that it is given, apart an x dependent correction that is bounded for large t, by the difference of two independent Poisson processes with parameters λ0 and λ1
Trace distance ergodicity for quantum Markov semigroups
Full text linkWe discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral decomposition of the Lindblad generator. We then apply this criterion to the bosonic and fermionic Ornstein-Uhlenbeck semigroups and to a family of quantum Markov semigroups parametrized by semisimple Lie algebras and their irreducible representations, in which the Lindblad generator is given by the adjoint action of the Casimir element
A Note on the physical meaning of averaged amplitudes and intensities in parametric instabilities driven by a stochastic pump
No full textThe problem of the physical meaning of averaged amplitudes and intensities in parametric instabilities driven by a stochastic pump is addressed. The physical meaning of an averaged quantity is understood as its ability to yield a good estimate of the median of this quantity. In the limit of a purely temporal problem with a Gaussian white noise pump, rigorous results can be obtained which show that no physical meaning can be bestowed a priori either on the averaged amplitudes or on the averaged intensities. A conjecture is proposed which enlarges this result to the general case. The statistics of the wave amplitudes is obtained by constructing a tractable weak solution to the parametric problem
Propagation of chaos for a general balls into bins dynamics
Full text linkConsider N balls initially placed in L bins. At each time step take a ball from each non- empty bin and randomly reassign all the balls into the bins. We call this finite Markov chain General Repeated Balls into Bins process. It is a discrete time conservative interacting particles system with parallel updates. Assuming a quantitative chaotic condition on the reassignment rule we prove a quantitative propagation of chaos for this model. We furthermore study some equilibrium properties of the limiting nonlinear process
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