The Bennett acceptance ratio method (BAR) is an algorithm for estimating the difference in free energy between two systems (usually the systems will be simulated on the computer).
It was suggested by
Charles H. Bennett in 1976.
Preliminaries
Take a system in a certain super (i.e. Gibbs) state. By performing a
Metropolis Monte Carlo walk it is possible to sample the landscape of states that the system moves between, using the equation
:
where Δ''U'' = ''U''(State
''y'') − ''U''(State
''x'') is the difference in potential energy, β = 1/''kT'' (''T'' is the temperature in
kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
s, while ''k'' is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
), and
is the Metropolis function.
The resulting states are then sampled according to the
Boltzmann distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability th ...
of the super state at temperature ''T''.
Alternatively, if the system is dynamically simulated in the
canonical ensemble
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the hea ...
(also called the ''NVT'' ensemble), the resulting states along the simulated trajectory are likewise distributed.
Averaging along the trajectory (in either formulation) is denoted by angle brackets
.
Suppose that two super states of interest, A and B, are given. We assume that they have a common configuration space, i.e., they share all of their micro states, but the energies associated to these (and hence the probabilities) differ because of a change in some parameter (such as the strength of a certain interaction).
The basic question to be addressed is, then, how can the
Helmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz en ...
change (Δ''F'' = ''F''
B − ''F''
A) on moving between the two super states be calculated from sampling in both ensembles? The kinetic energy part in the free energy is equal between states so can be ignored. Also the
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature an ...
corresponds to the ''NpT'' ensemble.
The general case
Bennett shows that for every function ''f'' satisfying the condition
(which is essentially the
detailed balance The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reve ...
condition), and for every energy offset ''C'', one has the exact relationship
:
where ''U''
A and ''U''
B are the potential energies of the same configurations, calculated using potential function A (when the system is in superstate A) and potential function B (when the system is in the superstate B) respectively.
The basic case
Substituting for ''f'' the Metropolis function defined above (which satisfies the detailed balance condition), and setting ''C'' to zero, gives
:
The advantage of this formulation (apart from its simplicity) is that it can be computed without performing two simulations, one in each specific ensemble. Indeed, it is possible to define an extra kind of "potential switching" Metropolis trial move (taken every fixed number of steps), such that the single sampling from the "mixed" ensemble suffices for the computation.
The most efficient case
Bennett explores which specific expression for Δ''F'' is the most efficient, in the sense of yielding the smallest standard error for a given simulation time. He shows that the optimal choice is to take
#
, which is essentially the
Fermi–Dirac distribution (satisfying indeed the detailed balance condition).
#
. This value, of course, is not known (it is exactly what one is trying to compute), but it can be approximately chosen in a self-consistent manner.
Some assumptions needed for the efficiency are the following:
# The densities of the two super states (in their common configuration space) should have a large overlap. Otherwise, a chain of super states between A and B may be needed, such that the overlap of each two consecutive super states is adequate.
# The sample size should be large. In particular, as successive states are correlated, the simulation time should be much larger than the correlation time.
# The cost of simulating both ensembles should be approximately equal - and then, in fact, the system is sampled roughly equally in both super states. Otherwise, the optimal expression for ''C'' is modified, and the sampling should devote equal times (rather than equal number of time steps) to the two ensembles.
Multistate Bennett acceptance ratio
The multistate Bennett acceptance ratio (MBAR) is a generalization of the Bennett acceptance ratio that calculates the (relative) free energies of several multi states. It essentially reduces to the BAR method when only two super states are involved.
Relation to other methods
The perturbation theory method
This method, also called
Free energy perturbation
Free energy perturbation (FEP) is a method based on statistical mechanics that is used in computational chemistry for computing free energy differences from molecular dynamics or Metropolis Monte Carlo simulations.
The FEP method was introduce ...
(or FEP), involves sampling from state A only. It requires that all the high probability configurations of super state B are contained in high probability configurations of super state A, which is a much more stringent requirement than the overlap condition stated above.
The exact (infinite order) result
:
or
:
This exact result can be obtained from the general BAR method, using (for example) the Metropolis function, in the limit
. Indeed, in that case, the denominator of the general case expression above tends to 1, while the numerator tends to
.
A direct derivation from the definitions is more straightforward, though.
The second order (approximate) result
Assuming that
and Taylor expanding the second exact perturbation theory expression to the second order, one gets the approximation
:
Note that the first term is the expected value of the energy difference, while the second is essentially its variance.
The first order inequalities
Using the convexity of the log function appearing in the exact perturbation analysis result, together with
Jensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier pr ...
, gives an inequality in the linear level; combined with the analogous result for the B ensemble one gets the following version of the
Gibbs-Bogoliubov inequality:
:
Note that the inequality agrees with the negative sign of the coefficient of the (positive) variance term in the second order result.
The thermodynamic integration method
writing the potential energy as depending on a continuous parameter,
one has the exact result
This can either be directly verified from definitions or seen from the limit of the above Gibbs-Bogoliubov inequalities when
.
we can therefore write
:
which is the
thermodynamic integration (or TI) result. It can be approximated by dividing the range between states A and B into many values of λ at which the expectation value is estimated, and performing numerical integration.
Implementation
The Bennett acceptance ratio method is implemented in modern
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
systems, such as
Gromacs.
Python-based code for MBAR and BAR is available for download a
See also
*
Parallel tempering
References
{{Reflist, refs=
[ Charles H. Bennett (1976) Efficient estimation of free energy differences from Monte Carlo data. ''Journal of Computational Physics'' 22 : 245–26]
External links
Bennett Acceptance Ratiofrom AlchemistryWiki.
Multistate Bennett Acceptance Ratiofrom AlchemistryWiki.
Weighted Histogram Analysis Method (MBAR being the unbinned case)from AlchemistryWiki.
Thermodynamics
Statistical mechanics