573 research outputs found
Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment
No full textA quantum system (with Hilbert space H1) entangled with its environment (with Hilbert space H2) is usually not attributed a wave function but only a reduced density matrix ρ1. Nevertheless, there is a precise way of attributing to it a random wave function ψ1, called its conditional wave function, whose probability distribution μ1 depends on the entangled wave function ψ∈H1⊗H2 in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of H2 but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about μ1, e.g., that if the environment is sufficiently large then for every orthonormal basis of H2, most entangled states ψ with given reduced density matrix ρ1 are such that μ1 is close to one of the so-called GAP (Gaussian adjusted projected) measures, GAP(ρ1). We also show that, for most entangled states ψ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval [E,E+δE]) and most orthonormal bases of H2, μ1 is close to GAP(tr2ρmc) with ρmc the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then μ1 is close to GAP(ρβ) with ρβ the canonical density matrix on H1 at inverse temperature β=β(E). This provides the mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that GAP measures describe the thermal equilibrium distribution of the wave function.Peer reviewe
Large deviations for a stochastic model of heat flow
No full textWe investigate a one-dimensional chain of 2N harmonic oscillators in which
neighboring sites have their energies redistributed randomly. The sites −N and
N are in contact with thermal reservoirs at different temperature τ− and τ+.
Kipnis et al. (J. Statist. Phys., 27:65–74 (1982).) proved that this model satisfies
Fourier’s law and that in the hydrodynamical scaling limit, when N→∞, the
stationary state has a linear energy density profile ̄ θ(u), u ∈ [−1, 1]. We derive
the large deviation function S(θ(u)) for the probability of finding, in the stationary
state, a profile θ(u) different from ̄ θ(u). The function S(θ) has striking
similarities to, but also large differences from, the corresponding one of the
symmetric exclusion process. Like the latter it is nonlocal and satisfies a variational
equation. Unlike the latter it is not convex and the Gaussian normal
fluctuations are enhanced rather than suppressed compared to the local equilibrium
state. We also briefly discuss more general models and find the features
common in these two and other models whose S(θ) is know
Checkerboards, stripes, and corner energies in spin models with competing interactions
No full textWe study the zero-temperature phase diagram of Ising spin systems in two dimensions in the presence of competing interactions: long-range antiferromagnetic and nearest-neighbor ferromagnetic of strength J. We first introduce the notion of a “corner energy,” which shows, when the antiferromagnetic interaction decays faster than the fourth power of the distance, that a striped state is favored with respect to a checkerboard state when J is close to Jc, the transition to the ferromagnetic state, i.e., when the length scales of the uniformly magnetized domains become large. Next, we perform detailed analytic computations on the energies of the striped and checkerboard states in the cases of antiferromagnetic interactions with exponential decay and with power-law decay r−p, p>2, which depend on the Manhattan distance instead of the Euclidean distance. We prove that the striped phase is always favored compared to the checkerboard phase when the scale of the ground-state structure is very large. This happens for J≲Jc if p>3, and for J sufficiently large if 2<p⩽3. Many of our considerations involving rigorous bounds carry over to dimensions greater than two and to more general short-range ferromagnetic interactions
Local mean field models of uniform to nonuniform density (fluid-crystal) transitions
No full textWe investigate the existence of nontranslation invariant (periodic) density profiles, for systems interacting via translation invariant long-range potentials, as minimizers of local mean field free energy functionals. The existence of a second-order transition from a uniform to a nonuniform density at a specified temperature beta(-1)(0) is proven for a class of model systems
The blockage problem
No full textWe investigate the totally asymmetric exclusion process on Z, with the jump rate at
site i given by ri = 1 for i 6= 0, r0 = r. It is easy to see that the maximal stationary current
j(r) is nondecreasing in r and that j(r) = 1/4 for r ≥ 1; it is a long outstanding problem to
determine whether or not the critical value rc of r such that j(r) = 1/4 for r > rc is strictly
less than 1. Here we present a heuristic argument, based on the analysis of the first sixteen
terms in a formal power series expansion of j(r) obtained from finite volume systems, that
rc = 1 and that for r / 1, j(r) ≃ 1/4 − γ exp[−a/(1 − r)] with a ≈ 2. We also give some
new exact results about this system; in particular we prove that j(r) = Jmax(r), with
Jmax(r) the hydrodynamic maximal current defined by Sepp ̈al ̈ainen, and thus establish
continuity of j(r). Finally we describe a related exactly solvable model, a semi-infinite
system in which the site i = 0 is always occupied. For that system, r
s-i
c = 1/2 and the
analogue j
s-i(r) of j(r) satisfies j
s-i(r) = r(1 − r) for r ≤ r
s-i
c
; j
s-i(r) is the limit of finite
volume currents inside the curve |r(1 − r)| = 1/4 in the complex r plane and we suggest
that analogous behavior may hold for the original system
Ising models with long-range dipolar and short range ferromagnetic interactions
No full textWe study the ground state of a d-dimensional Ising model with both long-range (dipole-like) and nearest-neighbor ferromagnetic (FM) interactions. The long-range interaction is equal to r−p, p>d, while the FM interaction has strength J. If p>d+1 and J is large enough the ground state is FM, while if d1 the ground state has a series of transitions from an antiferromagnetic state of period 2 to 2h-periodic states of blocks of sizes h with alternating sign, the size h growing when the FM interaction strength J is increased (a generalization of this result to the case 0<p⩽1 is also discussed). In d⩾2 we prove, for d<p⩽d+1, that the dominant asymptotic behavior of the ground-state energy agrees for large J with that obtained from a periodic striped state conjectured to be the true ground state. The geometry of contours in the ground state is discussed
Translation Invariant Extensions of Finite Volume Measures
No full textWe investigate the following questions: Given a measure μΛ on configurations on a subset Λ of a lattice L, where a configuration is an element of Ω Λ for some fixed set Ω , does there exist a measure μ on configurations on all of L, invariant under some specified symmetry group of L, such that μΛ is its marginal on configurations on Λ ? When the answer is yes, what are the properties, e.g., the entropies, of such measures? Our primary focus is the case in which L= Zd and the symmetries are the translations. For the case in which Λ is an interval in Z we give a simple necessary and sufficient condition, local translation invariance (LTI), for extendibility. For LTI measures we construct extensions having maximal entropy, which we show are Gibbs measures; this construction extends to the case in which L is the Bethe lattice. On Z we also consider extensions supported on periodic configurations, which are analyzed using de Bruijn graphs and which include the extensions with minimal entropy. When Λ ⊂ Z is not an interval, or when Λ ⊂ Zd with d> 1 , the LTI condition is necessary but not sufficient for extendibility. For Zd with d> 1 , extendibility is in some sense undecidable
Realizability of point processes
No full textThere are various situations in which it is natural to ask whether a given collection of k functions, ρ j (r 1,...,r j ), j=1,...,k, defined on a set X, are the first k correlation functions of a point process on X. Here we describe some necessary and sufficient conditions on the ρ j 's for this to be true. Our primary examples are X=Rd , X=Zd , and X an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities ρ 1(r). Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when X is a finite set, the existence of a realizing Gibbs measure with k body potentials which maximizes the entropy among all realizing measures. We also investigate in detail a simple example in which a uniform density ρ and translation invariant ρ 2 are specified on Z; there is a gap between our best upper bound on possible values of ρ and the largest ρ for which realizability can be established. © 2007 Springer Science+Business Media, LLC
- …
