208 research outputs found
Investigation of a generalized Obukhov Model for Turbulence
We introduce a generalization of Obukhov's model [A.M. Obukhov, Adv. Geophys.
6, 113 (1959)] for the description of the joint position-velocity statistics of
a single fluid particle in fully developed turbulence. In the presented model
the velocity is assumed to undergo a continuous time random walk. This takes
into account long time correlations. As a consequence the evolution equation
for the joint position-velocity probability distribution is a Fokker-Planck
equation with a fractional time derivative. We determine the solution of this
equation in the form of an integral transform and derive a relation for
arbitrary single time moments. Analytical solutions for the joint probability
distribution and its moments are given.Comment: 10 page
Entropy production for velocity-dependent macroscopic forces: the problem of dissipation without fluctuations
In macroscopic systems, velocity-dependent phenomenological forces are
used to model friction, feedback devices or self-propulsion. Such forces
usually include a dissipative component which conceals the fast energy
exchanges with a thermostat at the environment temperature , ruled by a
microscopic Hamiltonian . The mapping - even if effective
for many purposes - may lead to applications of stochastic thermodynamics where
an fluctuating entropy production (FEP) is derived. An
enlightening example is offered by recent macroscopic experiments where
dissipation is dominated by solid-on-solid friction, typically modelled through
a deterministic Coulomb force . Through an adaptation of the microscopic
Prandtl-Tomlinson model for friction, we show how the FEP is dominated by the
heat released to the -thermostat, ignored by the macroscopic Coulomb model.
This problem, which haunts several studies in the literature, cannot be cured
by weighing the time-reversed trajectories with a different auxiliary dynamics:
it is only solved by a more accurate stochastic modelling of the thermostat
underlying dissipation.Comment: 6 pages, 3 figure
Weak Galilean invariance as a selection principle for coarse-grained diffusive models
Galilean invariance is a cornerstone of classical mechanics. It states that
for closed systems the equations of motion of the microscopic degrees of
freedom do not change under Galilean transformations to different inertial
frames. However, the description of real world systems usually requires
coarse-grained models integrating complex microscopic interactions
indistinguishably as friction and stochastic forces, which intrinsically
violate Galilean invariance. By studying the coarse-graining procedure in
different frames, we show that alternative rules -- denoted as "weak Galilean
invariance" -- need to be satisfied by these stochastic models. Our results
highlight that diffusive models in general can not be chosen arbitrarily based
on the agreement with data alone but have to satisfy weak Galilean invariance
for physical consistency
Nonequilibrium statistical mechanics of shear flow: invariant quantities and current relations
In modeling nonequilibrium systems one usually starts with a definition of
the microscopic dynamics, e.g., in terms of transition rates, and then derives
the resulting macroscopic behavior. We address the inverse question for a class
of steady state systems, namely complex fluids under continuous shear flow: how
does an externally imposed shear current affect the microscopic dynamics of the
fluid? The answer can be formulated in the form of invariant quantities, exact
relations for the transition rates in the nonequilibrium steady state, as
discussed in a recent letter [A. Baule and R. M. L. Evans, Phys. Rev. Lett.
101, 240601 (2008)]. Here, we present a more pedagogical account of the
invariant quantities and the theory underlying them, known as the
nonequilibrium counterpart to detailed balance (NCDB). Furthermore, we
investigate the relationship between the transition rates and the shear current
in the steady state. We show that a fluctuation relation of the
Gallavotti-Cohen type holds for systems satisfying NCDB.Comment: 24 pages, 11 figure
Stick-slip motion of solids with dry friction subject to random vibrations and an external field
We investigate a model for the dynamics of a solid object, which moves over a
randomly vibrating solid surface and is subject to a constant external force.
The dry friction between the two solids is modeled phenomenologically as being
proportional to the sign of the object's velocity relative to the surface, and
therefore shows a discontinuity at zero velocity. Using a path integral
approach, we derive analytical expressions for the transition probability of
the object's velocity and the stationary distribution of the work done on the
object due to the external force. From the latter distribution, we also derive
a fluctuation relation for the mechanical work fluctuations, which incorporates
the effect of the dry friction.Comment: v1: 23 pages, 9 figures; v2: Reference list corrected; v3: Published
version, typos corrected, references adde
Statistical mechanics far from equilibrium: prediction and test for a sheared system
We report the complete statistical treatment of a system of particles
interacting via Newtonian forces in continuous boundary-driven flow, far from
equilibrium. By numerically time-stepping the force-balance equations of a
model fluid we measure occupancies and transition rates in simulation. The
high-shear-rate simulation data verify the invariant quantities predicted by
our statistical theory, thus demonstrating that a class of non-equilibrium
steady states of matter, namely sheared complex fluids, is amenable to
statistical treatment from first principles.Comment: 4 pages plus a 3-page pdf supplemen
Validation of the Jarzynski relation for a system with strong thermal coupling: an isothermal ideal gas model
We revisit the paradigm of an ideal gas under isothermal conditions. A moving piston performs work on an ideal gas in a container that is strongly coupled to a heat reservoir. The thermal coupling is modeled by stochastic scattering at the boundaries. In contrast to recent studies of an adiabatic ideal gas with a piston [R.C. Lua and A.Y. Grosberg, J. Phys. Chem. B 109, 6805 (2005); I. Bena et al., Europhys. Lett. 71, 879 (2005)], the container and piston stay in contact with the heat bath during the work process. Under this condition the heat reservoir as well as the system depend on the work parameter lambda and microscopic reversibility is broken for a moving piston. Our model is thus not included in the class of systems for which the nonequilibrium work theorem has been derived rigorously either by Hamiltonian [C. Jarzynski, J. Stat. Mech. (2004) P09005] or stochastic methods [G.E. Crooks, J. Stat. Phys. 90, 1481 (1998)]. Nevertheless the validity of the nonequilibrium work theorem is confirmed both numerically for a wide range of parameter values and analytically in the limit of a very fast moving piston, i.e., in the far nonequilibrium regime
A fractional diffusion equation for two-point probability distributions of a continuous-time random walk
Continuous time random walks are non-Markovian stochastic processes, which
are only partly characterized by single-time probability distributions. We
derive a closed evolution equation for joint two-point probability density
functions of a subdiffusive continuous time random walk, which can be
considered as a generalization of the known single-time fractional diffusion
equation to two-time probability distributions. The solution of this
generalized diffusion equation is given as an integral transformation of the
probability distribution of an ordinary diffusion process, where the integral
kernel is generated by an inverse L\'evy stable process. Explicit expressions
for the two time moments of a diffusion process are given, which could be
readily compared with the ones determined from experiments.Comment: 6 pages, 1 figur
Anomalous Processes with General Waiting Times: Functionals and Multipoint Structure
Many transport processes in nature exhibit anomalous diffusive properties
with non-trivial scaling of the mean square displacement, e.g., diffusion of
cells or of biomolecules inside the cell nucleus, where typically a crossover
between different scaling regimes appears over time. Here, we investigate a
class of anomalous diffusion processes that is able to capture such complex
dynamics by virtue of a general waiting time distribution. We obtain a complete
characterization of such generalized anomalous processes, including their
functionals and multi-point structure, using a representation in terms of a
normal diffusive process plus a stochastic time change. In particular, we
derive analytical closed form expressions for the two-point correlation
functions, which can be readily compared with experimental data.Comment: Accepted in Phys. Rev. Let
Exact power spectra of Brownian motion with solid friction
We study a Langevin equation describing the Brownian motion of an object
subjected to a viscous drag, an external constant force, and a solid friction
force of the Coulomb type. In a previous work [H. Touchette, E. Van der
Straeten, W. Just, J. Phys. A: Math. Theor. 43, 445002, 2010], we have
presented the exact solution of the velocity propagator of this equation based
on a spectral decomposition of the corresponding Fokker-Planck equation. Here,
we present an alternative, exact solution based on the Laplace transform of
this equation, which has the advantage of being expressed in closed form. From
this solution, we also obtain closed-form expressions for the Laplace transform
of the velocity autocorrelation function and for the power spectrum, i.e., the
Fourier transform of the autocorrelation function. The behavior of the power
spectrum as a function of the dry friction force and external forcing shows a
clear crossover between stick and slip regimes known to occur in the presence
of solid friction.Comment: v1: 14 pages, 5 figures; v2: new figures, some text added, typos
correcte
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