24 research outputs found
Computing probabilities of very rare events for Langevin processes: a new method based on importance sampling
Langevin equations are used to model many processes of physical interest,
including low-energy nuclear collisions. In this paper we develop a general
method for computing probabilities of very rare events (e.g. small fusion
cross-sections) for processes described by Langevin dynamics. As we demonstrate
with numerical examples as well as an exactly solvable model, our method can
converge to the desired answer at a rate which is orders of magnitude faster
than that achieved with direct simulations of the process in question.Comment: 18 pages + 7 figures, to appear in Nucl.Phys.
Work distribution for the driven harmonic oscillator with time-dependent strength: Exact solution and slow driving
We study the work distribution of a single particle moving in a harmonic
oscillator with time-dependent strength. This simple system has a non-Gaussian
work distribution with exponential tails. The time evolution of the
corresponding moment generating function is given by two coupled ordinary
differential equations that are solved numerically. Based on this result we
study the behavior of the work distribution in the limit of slow but finite
driving and show that it approaches a Gaussian distribution arbitrarily well
Probability distributions of the work in the 2D-Ising model
Probability distributions of the magnetic work are computed for the 2D Ising
model by means of Monte Carlo simulations. The system is first prepared at
equilibrium for three temperatures below, at and above the critical point. A
magnetic field is then applied and grown linearly at different rates.
Probability distributions of the work are stored and free energy differences
computed using the Jarzynski equality. Consistency is checked and the dynamics
of the system is analyzed. Free energies and dissipated works are reproduced
with simple models. The critical exponent is estimated in an usual
manner.Comment: 12 pages, 6 figures. Comments are welcom
Fluctuation relations for heat engines in time-periodic steady states
A fluctuation relation for heat engines (FRHE) has been derived recently. In
the beginning, the system is in contact with the cooler bath. The system is
then coupled to the hotter bath and external parameters are changed cyclically,
eventually bringing the system back to its initial state, once the coupling
with the hot bath is switched off. In this work, we lift the condition of
initial thermal equilibrium and derive a new fluctuation relation for the
central system (heat engine) being in a time-periodic steady state (TPSS).
Carnot's inequality for classical thermodynamics follows as a direct
consequence of this fluctuation theorem even in TPSS. For the special cases of
the absence of hot bath and no extraction of work, we obtain the integral
fluctuation theorem for total entropy and the generalized exchange fluctuation
theorem, respectively. Recently microsized heat engines have been realized
experimentally in the TPSS. We numerically simulate the same model and verify
our proposed theorems.Comment: 9 page
Estimate of the free energy difference in mechanical systems from work fluctuations: experiments and models
The work fluctuations of an oscillator in contact with a heat reservoir and
driven out of equilibrium by an external force are studied experimentally. The
oscillator dynamics is modeled by a Langevin equation. We find both
experimentally and theoretically that, if the driving force does not change the
equilibrium properties of the thermal fluctuations of this mechanical system,
the free energy difference between two equilibrium states can be
exactly computed using the Jarzynski equality (JE) and the Crooks relation (CR)
\cite{jarzynski1, crooks1, jarzynski2}, independently of the time scale and
amplitude of the driving force. The applicability limits for the JE and CR at
very large driving forces are discussed. Finally, when the work fluctuations
are Gaussian, we propose an alternative empirical method to compute
which can be safely applied, even in cases where the JE and CR might not hold.
The results of this paper are useful to compute in complex systems
such as the biological ones.Comment: submitted to Journal of Statistical Mechanics: Theory and experimen
Diffusion over a saddle with a Langevin equation
The diffusion problem over a saddle is studied using a multi-dimensional
Langevin equation. An analytical solution is derived for a quadratic potential
and the probability to pass over the barrier deduced. A very simple solution is
given for the one dimension problem and a general scheme is shown for higher
dimensions.Comment: 13 pages, use revTeX, to appear in Phys. Rev. E6
Feynman's ratchet and pawl: an exactly solvable model
We introduce a simple, discrete model of Feynman's ratchet and pawl,
operating between two heat reservoirs. We solve exactly for the steady-state
directed motion and heat flows produced, first in the absence and then in the
presence of an external load. We show that the model can act both as a heat
engine and as a refrigerator. We finally investigate the behavior of the system
near equilibrium, and use our model to confirm general predictions based on
linear response theory.Comment: 19 pages + 10 figures; somewhat tighter presentatio
Efficient Dynamic Importance Sampling of Rare Events in One Dimension
Exploiting stochastic path integral theory, we obtain \emph{by simulation}
substantial gains in efficiency for the computation of reaction rates in
one-dimensional, bistable, overdamped stochastic systems. Using a well-defined
measure of efficiency, we compare implementations of ``Dynamic Importance
Sampling'' (DIMS) methods to unbiased simulation. The best DIMS algorithms are
shown to increase efficiency by factors of approximately 20 for a
barrier height and 300 for , compared to unbiased simulation. The
gains result from close emulation of natural (unbiased), instanton-like
crossing events with artificially decreased waiting times between events that
are corrected for in rate calculations. The artificial crossing events are
generated using the closed-form solution to the most probable crossing event
described by the Onsager-Machlup action. While the best biasing methods require
the second derivative of the potential (resulting from the ``Jacobian'' term in
the action, which is discussed at length), algorithms employing solely the
first derivative do nearly as well. We discuss the importance of
one-dimensional models to larger systems, and suggest extensions to
higher-dimensional systems.Comment: version to be published in Phys. Rev.
Work and heat fluctuations in two-state systems: a trajectory thermodynamics formalism
Two-state models provide phenomenological descriptions of many different
systems, ranging from physics to chemistry and biology. We investigate work
fluctuations in an ensemble of two-state systems driven out of equilibrium
under the action of an external perturbation. We calculate the probability
density P(W) that a work equal to W is exerted upon the system along a given
non-equilibrium trajectory and introduce a trajectory thermodynamics formalism
to quantify work fluctuations in the large-size limit. We then define a
trajectory entropy S(W) that counts the number of non-equilibrium trajectories
P(W)=exp(S(W)/kT) with work equal to W. A trajectory free-energy F(W) can also
be defined, which has a minimum at a value of the work that has to be
efficiently sampled to quantitatively test the Jarzynski equality. Within this
formalism a Lagrange multiplier is also introduced, the inverse of which plays
the role of a trajectory temperature. Our solution for P(W) exactly satisfies
the fluctuation theorem by Crooks and allows us to investigate
heat-fluctuations for a protocol that is invariant under time reversal. The
heat distribution is then characterized by a Gaussian component (describing
small and frequent heat exchange events) and exponential tails (describing the
statistics of large deviations and rare events). For the latter, the width of
the exponential tails is related to the aforementioned trajectory temperature.
Finite-size effects to the large-N theory and the recovery of work
distributions for finite N are also discussed. Finally, we pay particular
attention to the case of magnetic nanoparticle systems under the action of a
magnetic field H where work and heat fluctuations are predicted to be
observable in ramping experiments in micro-SQUIDs.Comment: 28 pages, 14 figures (Latex