4,113 research outputs found
Density profiles, dynamics, and condensation in the ZRP conditioned on an atypical current
We study the asymmetric zero-range process (ZRP) with L sites and open
boundaries, conditioned to carry an atypical current. Using a generalized Doob
h-transform we compute explicitly the transition rates of an effective process
for which the conditioned dynamics are typical. This effective process is a
zero-range process with renormalized hopping rates, which are space dependent
even when the original rates are constant. This leads to non-trivial density
profiles in the steady state of the conditioned dynamics, and, under generic
conditions on the jump rates of the unconditioned ZRP, to an intriguing
supercritical bulk region where condensates can grow. These results provide a
microscopic perspective on macroscopic fluctuation theory (MFT) for the weakly
asymmetric case: It turns out that the predictions of MFT remain valid in the
non-rigorous limit of finite asymmetry. In addition, the microscopic results
yield the correct scaling factor for the asymmetry that MFT cannot predict.Comment: 26 pages, 4 figure
Robustness of spontaneous symmetry breaking in a bridge model
A simple two-species asymmetric exclusion model in one dimension with bulk
and boundary exchanges of particles is investigated for the existence of
spontaneous symmetry breaking. The model is a generalization of the bridge
model for which earlier studies have confirmed the existence of symmetry-broken
phases, and the motivation here is to check the robustness of the observed
symmetry breaking with respect to additional dynamical moves, in particular,
the boundary exchange of the two species of particles. Our analysis, based on
general considerations, mean-field approximation and numerical simulations,
shows that the symmetry breaking in the bridge model is sustained for a range
of values of the boundary exchange rate. Moreover, the mechanism through which
symmetry is broken is similar to that in the bridge model. Our analysis allows
us to plot the complete phase diagram of the model, demarcating regions of
symmetric and symmetry-broken phases.Comment: 26 pages, 12 figures, v2: minor changes with an added appendix,
published versio
Diffusion in a logarithmic potential: scaling and selection in the approach to equilibrium
The equation which describes a particle diffusing in a logarithmic potential
arises in diverse physical problems such as momentum diffusion of atoms in
optical traps, condensation processes, and denaturation of DNA molecules. A
detailed study of the approach of such systems to equilibrium via a scaling
analysis is carried out, revealing three surprising features: (i) the solution
is given by two distinct scaling forms, corresponding to a diffusive (x ~
\sqrt{t}) and a subdiffusive (x >> \sqrt{t}) length scales, respectively; (ii)
the scaling exponents and scaling functions corresponding to both regimes are
selected by the initial condition; and (iii) this dependence on the initial
condition manifests a "phase transition" from a regime in which the scaling
solution depends on the initial condition to a regime in which it is
independent of it. The selection mechanism which is found has many similarities
to the marginal stability mechanism which has been widely studied in the
context of fronts propagating into unstable states. The general scaling forms
are presented and their practical and theoretical applications are discussed.Comment: 42 page
Set-Theoretic Geology
A ground of the universe V is a transitive proper class W subset V, such that
W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G]
for some W-generic filter G subset P in W . The model V satisfies the ground
axiom GA if there are no such W properly contained in V . The model W is a
bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle
of V is the intersection of all grounds of V . The generic mantle of V is the
intersection of all grounds of all set-forcing extensions of V . The generic
HOD, written gHOD, is the intersection of all HODs of all set-forcing
extensions. The generic HOD is always a model of ZFC, and the generic mantle is
always a model of ZF. Every model of ZFC is the mantle and generic mantle of
another model of ZFC. We prove this theorem while also controlling the HOD of
the final model, as well as the generic HOD. Iteratively taking the mantle
penetrates down through the inner mantles to what we call the outer core, what
remains when all outer layers of forcing have been stripped away. Many
fundamental questions remain open.Comment: 44 pages; commentary concerning this article can be made at
http://jdh.hamkins.org/set-theoreticgeology
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