108 research outputs found
A -uniform quantitative Tanaka's theorem for the conservative Kac's -particle system with Maxwell molecules
This paper considers the space homogenous Boltzmann equation with Maxwell
molecules and arbitrary angular distribution. Following Kac's program, emphasis
is laid on the the associated conservative Kac's stochastic -particle
system, a Markov process with binary collisions conserving energy and total
momentum. An explicit Markov coupling (a probabilistic, Markovian coupling of
two copies of the process) is constructed, using simultaneous collisions, and
parallel coupling of each binary random collision on the sphere of collisional
directions. The euclidean distance between the two coupled systems is almost
surely decreasing with respect to time, and the associated quadratic coupling
creation (the time variation of the averaged squared coupling distance) is
computed explicitly. Then, a family (indexed by ) of -uniform
''weak'' coupling / coupling creation inequalities are proven, that leads to a
-uniform power law trend to equilibrium of order , with constants depending on moments of the velocity
distributions strictly greater than . The case of order
moment is treated explicitly, achieving Kac's program without any chaos
propagation analysis. Finally, two counter-examples are suggested indicating
that the method: (i) requires the dependance on -moments, and (ii) cannot
provide contractivity in quadratic Wasserstein distance in any case.Comment: arXiv admin note: text overlap with arXiv:1312.225
Scalable and Quasi-Contractive Markov Coupling of Maxwell Collision
This paper considers space homogenous Boltzmann kinetic equations in
dimension with Maxwell collisions (and without Grad's cut-off). An explicit
Markov coupling of the associated conservative (Nanbu) stochastic -particle
system is constructed, using plain parallel coupling of isotropic random walks
on the sphere of two-body collisional directions. The resulting coupling is
almost surely decreasing, and the -coupling creation is computed
explicitly. Some quasi-contractive and uniform in coupling / coupling
creation inequalities are then proved, relying on -moments () of velocity distributions; upon -uniform propagation of moments of the
particle system, it yields a -scalable -power law trend to
equilibrium. The latter are based on an original sharp inequality, which bounds
from above the coupling distance of two centered and normalized random
variables in , with the average square parallelogram area spanned
by , denoting an independent copy. Two
counter-examples proving the necessity of the dependance on -moments and
the impossibility of strict contractivity are provided. The paper, (mostly)
self-contained, does not require any propagation of chaos property and uses
only elementary tools.Comment: 29 page
On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes.
This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of fermion groundstates, defined by the bottom eigenelements of the Schrödinger operator of a fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that the shape derivative of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds
Analysis of Adaptive Multilevel Splitting algorithms in an idealized case
The Adaptive Multilevel Splitting algorithm is a very powerful and versatile
method to estimate rare events probabilities. It is an iterative procedure on
an interacting particle system, where at each step, the less well-adapted
particles among are killed while new better adapted particles are
resampled according to a conditional law. We analyze the algorithm in the
idealized setting of an exact resampling and prove that the estimator of the
rare event probability is unbiased whatever . We also obtain a precise
asymptotic expansion for the variance of the estimator and the cost of the
algorithm in the large limit, for a fixed
Long-time convergence of an Adaptive Biasing Force method
We propose a proof of convergence of an adaptive method used in molecular
dynamics to compute free energy profiles. Mathematically, it amounts to
studying the long-time behavior of a stochastic process which satisfies a
non-linear stochastic differential equation, where the drift depends on
conditional expectations of some functionals of the process. We use entropy
techniques to prove exponential convergence to the stationary state
Computation of free energy profiles with parallel adaptive dynamics
We propose a formulation of adaptive computation of free energy differences,
in the ABF or nonequilibrium metadynamics spirit, using conditional
distributions of samples of configurations which evolve in time. This allows to
present a truly unifying framework for these methods, and to prove convergence
results for certain classes of algorithms. From a numerical viewpoint, a
parallel implementation of these methods is very natural, the replicas
interacting through the reconstructed free energy. We show how to improve this
parallel implementation by resorting to some selection mechanism on the
replicas. This is illustrated by computations on a model system of
conformational changes.Comment: 4 pages, 1 Figur
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