398 research outputs found
Entropy and density of states from isoenergetic nonequilibrium processes
Two identities in statistical mechanics involving entropy differences (or
ratios of density of states) at constant energy are derived. The first provides
a nontrivial extension of the Jarzynski equality to the microcanonical ensemble
[C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)], which can be seen as a
``fast-switching'' version of the adiabatic switching method for computing
entropies [M. Watanabe, W. P. Reinhardt, Phys. Rev. Lett. 65, 3301 (1990)]. The
second is a thermodynamic integration formula analogous to a well-known
expression for free energies, and follows after taking the quasistatic limit of
the first. Both identities can be conveniently used in conjunction with a
scaling relation (herein derived) that allows one to extrapolate measurements
taken at a single energy to a wide range of energy values. Practical aspects of
these identities in the context of numerical simulations are discussed.Comment: 5 pages, no figure
Phase transition in the Jarzynski estimator of free energy differences
The transition between a regime in which thermodynamic relations apply only
to ensembles of small systems coupled to a large environment and a regime in
which they can be used to characterize individual macroscopic systems is
analyzed in terms of the change in behavior of the Jarzynski estimator of
equilibrium free energy differences from nonequilibrium work measurements.
Given a fixed number of measurements, the Jarzynski estimator is unbiased for
sufficiently small systems. In these systems, the directionality of time is
poorly defined and configurations that dominate the empirical average, but
which are in fact typical of the reverse process, are sufficiently well
sampled. As the system size increases the arrow of time becomes better defined.
The dominant atypical fluctuations become rare and eventually cannot be sampled
with the limited resources that are available. Asymptotically, only typical
work values are measured. The Jarzynski estimator becomes maximally biased and
approaches the exponential of minus the average work, which is the result that
is expected from standard macroscopic thermodynamics. In the proper scaling
limit, this regime change can be described in terms of a phase transition in
variants of the random energy model (REM). This correspondence is explicitly
demonstrated in several examples of physical interest: near-equilibrium
processes in which the work distribution is Gaussian, the sudden compression of
an ideal gas and adiabatic quasi-static volume changes in a dilute real gas.Comment: 29 pages, 5 figures, accepted for publication in Physical Review E
(2012
Equilibrium binding energies from fluctuation theorems and force spectroscopy simulations
Brownian dynamics simulations are used to study the detachment of a particle
from a substrate. Although the model is simple and generic, we attempt to map
its energy, length and time scales onto a specific experimental system, namely
a bead that is weakly bound to a cell and then removed by an optical tweezer.
The external driving force arises from the combined optical tweezer and
substrate potentials, and thermal fluctuations are taken into account by a
Brownian force. The Jarzynski equality and Crooks' fluctuation theorem are
applied to obtain the equilibrium free energy difference between the final and
initial states. To this end, we sample non--equilibrium work trajectories for
various tweezer pulling rates. We argue that this methodology should also be
feasible experimentally for the envisioned system. Furthermore, we outline how
the measurement of a whole free energy profile would allow the experimentalist
to retrieve the unknown substrate potential by means of a suitable
deconvolution. The influence of the pulling rate on the accuracy of the results
is investigated, and umbrella sampling is used to obtain the equilibrium
probability of particle escape for a variety of trap potentials.Comment: 21 pages, 11 figures, To appear in Soft Matte
Free energy of formation of clusters of sulphuric acid and water molecules determined by guided disassembly
We evaluate the grand potential of a cluster of two molecular species,
equivalent to its free energy of formation from a binary vapour phase, using a
nonequilibrium molecular dynamics technique where guide particles, each
tethered to a molecule by a harmonic force, move apart to disassemble a cluster
into its components. The mechanical work performed in an ensemble of
trajectories is analysed using the Jarzynski equality to obtain a free energy
of disassembly, a contribution to the cluster grand potential. We study
clusters of sulphuric acid and water at 300 K, using a classical interaction
scheme, and contrast two modes of guided disassembly. In one, the cluster is
broken apart through simple pulling by the guide particles, but we find the
trajectories tend to be mechanically irreversible. In the second approach, the
guide motion and strength of tethering are modified in a way that prises the
cluster apart, a procedure that seems more reversible. We construct a surface
representing the cluster grand potential, and identify a critical cluster for
droplet nucleation under given vapour conditions. We compare the equilibrium
populations of clusters with calculations reported by Henschel et al. [J. Phys.
Chem. A 118, 2599 (2014)] based on optimised quantum chemical structures
Density-Dependent Analysis of Nonequilibrium Paths Improves Free Energy Estimates II. A Feynman-Kac Formalism
The nonequilibrium fluctuation theorems have paved the way for estimating
equilibrium thermodynamic properties, such as free energy differences, using
trajectories from driven nonequilibrium processes. While many statistical
estimators may be derived from these identities, some are more efficient than
others. It has recently been suggested that trajectories sampled using a
particular time-dependent protocol for perturbing the Hamiltonian may be
analyzed with another one. Choosing an analysis protocol based on the
nonequilibrium density was empirically demonstrated to reduce the variance and
bias of free energy estimates. Here, we present an alternate mathematical
formalism for protocol postprocessing based on the Feynmac-Kac theorem. The
estimator that results from this formalism is demonstrated on a few
low-dimensional model systems. It is found to have reduced bias compared to
both the standard form of Jarzynski's equality and the previous protocol
postprocessing formalism.Comment: 21 pages, 5 figure
Work measurement as a generalized quantum measurement
We present a new method to measure the work performed on a driven quantum
system and to sample its probability distribution . The method is based
on a simple fact that remained unnoticed until now: Work on a quantum system
can be measured by performing a generalized quantum measurement at a single
time. Such measurement, which technically speaking is denoted as a POVM
(positive operator valued measure) reduces to an ordinary projective
measurement on an enlarged system. This observation not only demystifies work
measurement but also suggests a new quantum algorithm to efficiently sample the
distribution . This can be used, in combination with fluctuation
theorems, to estimate free energies of quantum states on a quantum computer.Comment: 4 page
Calculation of absolute free energy of binding for theophylline and its analogs to RNA aptamer using nonequilibrium work values
The massively parallel computation of absolute binding free energy with a
well-equilibrated system (MP-CAFEE) has been developed [H. Fujitani, Y. Tanida,
M. Ito, G. Jayachandran, C. D. Snow, M. R. Shirts, E. J. Sorin, and V. S.
Pande, J. Chem. Phys. , 084108 (2005)]. As an application, we
perform the binding affinity calculations of six theophylline-related ligands
with RNA aptamer. Basically, our method is applicable when using many compute
nodes to accelerate simulations, thus a parallel computing system is also
developed. To further reduce the computational cost, the adequate non-uniform
intervals of coupling constant , connecting two equilibrium states,
namely bound and unbound, are determined. The absolute binding energies thus obtained have effective linear relation between the computed and
experimental values. If the results of two other different methods are
compared, thermodynamic integration (TI) and molecular mechanics
Poisson-Boltzmann surface area (MM-PBSA) by the paper of Gouda [H.
Gouda, I. D. Kuntz, D. A. Case, and P. A. Kollman, Biopolymers , 16
(2003)], the predictive accuracy of the relative values is
almost comparable to that of TI: the correlation coefficients (R) obtained are
0.99 (this work), 0.97 (TI), and 0.78 (MM-PBSA). On absolute binding energies
meanwhile, a constant energy shift of -7 kcal/mol against the
experimental values is evident. To solve this problem, several presumable
reasons are investigated.Comment: 23 pages including 6 figure
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