Jeremy Clark

Bio: Jeremy Clark is an academic researcher from University of Mississippi. The author has contributed to research in topics: Test particle & Brownian motion. The author has an hindex of 5, co-authored 34 publications receiving 108 citations. Previous affiliations of Jeremy Clark include Michigan State University & Katholieke Universiteit Leuven.

Diffusive behavior from a quantum master equation

Abstract: We study a general class of translation invariant quantum Markov evolutions for a particle on Zd. The evolution consists of free flow, interrupted by scattering events. We assume spatial locality of the scattering events and exponentially fast relaxation of the momentum distribution. It is shown that the particle position diffuses in the long time limit. This generalizes standard results about central limit theorems for classical (non-quantum) Markov processes.

Diffusive behavior from a quantum master equation

Abstract: We study a general class of translation invariant quantum Markov evolutions for a particle on $\bbZ^d$. The evolution consists of free flow, interrupted by scattering events. We assume spatial locality of the scattering events and exponentially fast relaxation of the momentum distribution. It is shown that the particle position diffuses in the long time limit. This generalizes standard results about central limit theorems for classical (non-quantum) Markov processes.

High-temperature scaling limit for directed polymers on a hierarchical lattice with bond disorder

Abstract: Diamond “lattices” are sequences of recursively-defined graphs that provide a network of directed pathways between two fixed root nodes, A and B. The construction recipe for diamond graphs depends on a branching number $$b\in {\mathbb {N}}$$ and a segmenting number $$s\in {\mathbb {N}}$$ , for which a larger value of the ratio s / b intuitively corresponds to more opportunities for intersections between two randomly chosen paths. By attaching i.i.d. random variables to the bonds of the graphs, we construct a random Gibbs measure on the set of directed paths by assigning each path an “energy” through summing the random variables along the path. For the case $$b=s$$ , we propose a scaling regime in which the temperature grows along with the number of hierarchical layers of the graphs, and the partition function (the normalization factor of the Gibbs measure) appears to converge in law. We prove that all of the positive integer moments of the partition function converge in this limiting regime. The motivation of this work is to prove a functional limit theorem that is analogous to a previous result obtained in the $$b

Quantum linear Boltzmann equation

Abstract: We review the quantum version of the linear Boltzmann equation, which describes in a non-perturbative fashion, by means of scattering theory, how the quantum motion of a single test particle is affected by collisions with an ideal background gas. A heuristic derivation of this Lindblad master equation is presented, based on the requirement of translation-covariance and on the relation to the classical linear Boltzmann equation. After analyzing its general symmetry properties and the associated relaxation dynamics, we discuss a quantum Monte Carlo method for its numerical solution. We then review important limiting forms of the quantum linear Boltzmann equation, such as the case of quantum Brownian motion and pure collisional decoherence, as well as the application to matter wave optics. Finally, we point to the incorporation of quantum degeneracies and self-interactions in the gas by relating the equation to the dynamic structure factor of the ambient medium, and we provide an extension of the equation to include internal degrees of freedom.

On the Moments of the (2+1)-Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window

Abstract: The partition function of the directed polymer model on $${\mathbb {Z}}^{2+1}$$ undergoes a phase transition in a suitable continuum and weak disorder limit. In this paper, we focus on a window around the critical point. Exploiting local renewal theorems, we compute the limiting third moment of the space-averaged partition function, showing that it is uniformly bounded. This implies that the rescaled partition functions, viewed as a generalized random field on $${\mathbb {R}}^{2}$$, have non-trivial subsequential limits, and each such limit has the same explicit covariance structure. We obtain analogous results for the stochastic heat equation on $${\mathbb {R}}^2$$, extending previous work by Bertini and Cancrini (J Phys A Math Gen 31:615, 1998).

Diffusion of a Massive Quantum Particle Coupled to a Quasi-Free Thermal Medium

Abstract: We consider a heavy quantum particle with an internal degree of freedom moving on the d-dimensional lattice \({{\mathbb Z}^d}\) (e.g., a heavy atom with finitely many internal states). The particle is coupled to a thermal medium (bath) consisting of free relativistic bosons (photons or Goldstone modes) through an interaction of strength λ linear in creation and annihilation operators. The mass of the quantum particle is assumed to be of order λ−2, and we assume that the internal degree of freedom is coupled “effectively” to the thermal medium. We prove that the motion of the quantum particle is diffusive in d ≥ 4 and for λ small enough.

A Complete Bibliography of Publications in the Journal of Mathematical Physics: 2005{2009

Abstract: (2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p [282]. B−L [427]. α [216, 483]. α− z [322]. N = 2 [507]. D [222]. ẍ+ f(x)ẋ + g(x) = 0 [112, 111, 8, 5, 6]. Eτ,ηgl3 [148]. g [300]. κ [244]. L [205, 117]. L [164]. L∞ [368]. M [539]. P [27]. R [147]. Z2 [565]. Z n 2 [131]. Z2 × Z2 [25]. D(X) [166]. S(N) [110]. ∫l2 [154]. SU(2) [210]. N [196, 242]. O [386]. osp(1|2) [565]. p [113, 468]. p(x) [17]. q [437, 220, 92, 183]. R, d = 1, 2, 3 [279]. SDiff(S) [32]. σ [526]. SLq(2) [185]. SU(N) [490]. τ [440]. U(1) N [507]. Uq(sl 2) [185]. φ 2k [283]. φ [553]. φ4 [365]. ∨ [466]. VOA[M4] [33]. Z [550].