816 research outputs found
Duality and fluctuation relations for statistics of currents on cyclic graphs
We consider stochastic motion of a particle on a cyclic graph with
arbitrarily periodic time dependent kinetic rates. We demonstrate duality
relations for statistics of currents in this model and in its continuous
version of a diffusion in one dimension. Our duality relations are valid beyond
detailed balance constraints and lead to exact expressions that relate
statistics of currents induced by dual driving protocols. We also show that
previously known no-pumping theorems and some of the fluctuation relations,
when they are applied to cyclic graphs or to one dimensional diffusion, are
special consequences of our duality.Comment: 2 figure, 6 pages (In twocolumn). Accepted by JSTA
Moderate deviation principle for ergodic Markov chain. Lipschitz summands
For , we propose the MDP analysis for family where
be a homogeneous ergodic Markov chain, ,
when the spectrum of operator is continuous. The vector-valued function
is not assumed to be bounded but the Lipschitz continuity of is
required. The main helpful tools in our approach are Poisson's equation and
Stochastic Exponential; the first enables to replace the original family by
with a martingale while the second to avoid the
direct Laplace transform analysis
Ising models on power-law random graphs
We study a ferromagnetic Ising model on random graphs with a power-law degree
distribution and compute the thermodynamic limit of the pressure when the mean
degree is finite (degree exponent ), for which the random graph has a
tree-like structure. For this, we adapt and simplify an analysis by Dembo and
Montanari, which assumes finite variance degrees (). We further
identify the thermodynamic limits of various physical quantities, such as the
magnetization and the internal energy
Limiting dynamics for spherical models of spin glasses at high temperature
We analyze the coupled non-linear integro-differential equations whose
solutions is the thermodynamical limit of the empirical correlation and
response functions in the Langevin dynamics for spherical p-spin disordered
mean-field models. We provide a mathematically rigorous derivation of their FDT
solution (for the high temperature regime) and of certain key properties of
this solution, which are in agreement with earlier derivations based on
physical grounds
The Non--Ergodicity Threshold: Time Scale for Magnetic Reversal
We prove the existence of a non-ergodicity threshold for an anisotropic
classical Heisenberg model with all-to-all couplings. Below the threshold, the
energy surface is disconnected in two components with positive and negative
magnetizations respectively. Above, in a fully chaotic regime, magnetization
changes sign in a stochastic way and its behavior can be fully characterized by
an average magnetization reversal time. We show that statistical mechanics
predicts a phase--transition at an energy higher than the non-ergodicity
threshold. We assess the dynamical relevance of the latter for finite systems
through numerical simulations and analytical calculations. In particular, the
time scale for magnetic reversal diverges as a power law at the ergodicity
threshold with a size-dependent exponent, which could be a signature of the
phenomenon.Comment: 4 pages 4 figure
On the validity of entropy production principles for linear electrical circuits
We discuss the validity of close-to-equilibrium entropy production principles
in the context of linear electrical circuits. Both the minimum and the maximum
entropy production principle are understood within dynamical fluctuation
theory. The starting point are Langevin equations obtained by combining
Kirchoff's laws with a Johnson-Nyquist noise at each dissipative element in the
circuit. The main observation is that the fluctuation functional for time
averages, that can be read off from the path-space action, is in first order
around equilibrium given by an entropy production rate. That allows to
understand beyond the schemes of irreversible thermodynamics (1) the validity
of the least dissipation, the minimum entropy production, and the maximum
entropy production principles close to equilibrium; (2) the role of the
observables' parity under time-reversal and, in particular, the origin of
Landauer's counterexample (1975) from the fact that the fluctuating observable
there is odd under time-reversal; (3) the critical remark of Jaynes (1980)
concerning the apparent inappropriateness of entropy production principles in
temperature-inhomogeneous circuits.Comment: 19 pages, 1 fi
Fluctuations in Polymer Translocation
We investigate a model of chaperone-assisted polymer translocation through a
nanopore in a membrane. Translocation is driven by irreversible random
sequential absorption of chaperone proteins that bind to the polymer on one
side of the membrane. The proteins are larger than the pore and hence the
backward motion of the polymer is inhibited. This mechanism rectifies Brownian
fluctuations and results in an effective force that drags the polymer in a
preferred direction. The translocated polymer undergoes an effective biased
random walk and we compute the corresponding diffusion constant. Our methods
allow us to determine the large deviation function which, in addition to
velocity and diffusion constant, contains the entire statistics of the
translocated length.Comment: 20 pages, 6 figure
A Two-populations Ising model on diluted Random Graphs
We consider the Ising model for two interacting groups of spins embedded in
an Erd\"{o}s-R\'{e}nyi random graph. The critical properties of the system are
investigated by means of extensive Monte Carlo simulations. Our results
evidence the existence of a phase transition at a value of the inter-groups
interaction coupling which depends algebraically on the dilution of
the graph and on the relative width of the two populations, as explained by
means of scaling arguments. We also measure the critical exponents, which are
consistent with those of the Curie-Weiss model, hence suggesting a wide
robustness of the universality class.Comment: 11 pages, 4 figure
Random Planar Lattices and Integrated SuperBrownian Excursion
In this paper, a surprising connection is described between a specific brand
of random lattices, namely planar quadrangulations, and Aldous' Integrated
SuperBrownian Excursion (ISE). As a consequence, the radius r_n of a random
quadrangulation with n faces is shown to converge, up to scaling, to the width
r=R-L of the support of the one-dimensional ISE. More generally the
distribution of distances to a random vertex in a random quadrangulation is
described in its scaled limit by the random measure ISE shifted to set the
minimum of its support in zero.
The first combinatorial ingredient is an encoding of quadrangulations by
trees embedded in the positive half-line, reminiscent of Cori and Vauquelin's
well labelled trees. The second step relates these trees to embedded (discrete)
trees in the sense of Aldous, via the conjugation of tree principle, an
analogue for trees of Vervaat's construction of the Brownian excursion from the
bridge.
From probability theory, we need a new result of independent interest: the
weak convergence of the encoding of a random embedded plane tree by two contour
walks to the Brownian snake description of ISE.
Our results suggest the existence of a Continuum Random Map describing in
term of ISE the scaled limit of the dynamical triangulations considered in
two-dimensional pure quantum gravity.Comment: 44 pages, 22 figures. Slides and extended abstract version are
available at http://www.loria.fr/~schaeffe/Pub/Diameter/ and
http://www.iecn.u-nancy.fr/~chassain
Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise
We consider the family of stochastic partial differential equations indexed
by a parameter \eps\in(0,1], \begin{equation*} Lu^{\eps}(t,x) =
\eps\sigma(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*}
(t,x)\in(0,T]\times\Rd with suitable initial conditions. In this equation,
is a second-order partial differential operator with constant coefficients,
and are smooth functions and is a Gaussian noise, white
in time and with a stationary correlation in space. Let p^\eps_{t,x} denote
the density of the law of u^\eps(t,x) at a fixed point
(t,x)\in(0,T]\times\Rd. We study the existence of \lim_{\eps\downarrow 0}
\eps^2\log p^\eps_{t,x}(y) for a fixed . The results apply to a class
of stochastic wave equations with and to a class of stochastic
heat equations with .Comment: 39 pages. Will be published in the book " Stochastic Analysis and
Applications 2014. A volume in honour of Terry Lyons". Springer Verla
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