Free energy canonical ensemble

Expressions similar to those given above may be derived easily from partition functions in other ensembles.The choice of ensemble is very important in calculations of hydration entropy, enthalpy, and heat capacity, as discussed below. Many other quantities, including all free energies, are ensemble invariant, with the choice of ensemble affecting only system size dependence. For simplicity, the discussion here is therefore limited to the canonical ensemble except in such cases where a true ensemble dependence exists. 

In a canonical ensemble, the system is held at fixed (V, T, N). In a grand canonical ensemble the (V, T p) of the system are fixed. The change from to p as an independent variable is made by a Legendre transfomiation in which the dependent variable, the Flelmlioltz free energy, is replaced by the grand potential... 

The canonical ensemble corresponds to a system of fixed and V, able to exchange energy with a thennal bath at temperature T, which represents the effects of the surroundings. The thennodynamic potential is the Helmholtz free energy, and it is related to the partition fiinction follows ... 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Finding the best estimate of the free energy difference between two canonical ensembles on the same configurational space, for which finite samples are available, is a nontrivial problem. Charles Bennett [11] addressed this problem by developing the acceptance ratio estimator, which corresponds to the minimum statistical... 

The Helmholtz free energy, A, which is the thermodynamic potential, the natural independent variables of which are those of the canonical ensemble, can be expressed in terms of the partition function ... 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

As we will see further in the book, almost all methods for calculating free energies in chemical and biological problems by means of computer simulations of equilibrium systems rely on one of the three approaches that we have just outlined, or on their possible combination. These methods can be applied not only in the context of the canonical ensemble, but also in other ensembles. As will be discussed in Chap. 5, AA can be also estimated from nonequilibrium simulations, to such extent that FEP and TI methods can be considered as limiting cases of this approach. 

UU is the Hamiltonian difference (the perturbation) the angle brackets represent a canonical ensemble average performed on an equilibrated system designated by the subscript. Usually the kinetic component of the Hamiltonian is not included in the free energy calculation, and A- //, n can be replaced by the potential energy difference AU = Ui - U0. 

In spite of these potential concerns, the MEHMC method is expected to be a useful tool for many applications. One task for which it might be particularly well suited is to generate a canonical ensemble of representative configurations of a bio-molecular system quickly. Such an ensemble is needed, for example, to represent the initial conditions for the ensemble of trajectories used in fast-growth free energy perturbation methods such as the one suggested by Jarzynski s identity [104] (see also Chap. 5). 

There is considerable interest in the use of discretized path-integral simulations to calculate free energy differences or potentials of mean force using quantum statistical mechanics for many-body systems [140], The reader has already become familiar with this approach to simulating with classical systems in Chap. 7. The theoretical basis of such methods is the Feynmann path-integral representation [141], from which is derived the isomorphism between the equilibrium canonical ensemble of a... 

In the canonical example, we could estimate the free energy difference between two runs by examining the overlap in their probability distributions. Similarly, in the grand canonical ensemble, we can estimate the pressure difference between the two runs. If the conditions for run I arc f//1. V. > ) and for run 2 (po, VjK), then... 

Chapters 10 and 11 cover methods that apply to systems different from those discussed so far. First, the techniques for calculating chemical potentials in the grand canonical ensemble are discussed. Even though much of this chapter is focused on phase equilibria, the reader will discover that most of the methodology introduced in Chap. 3 can be easily adapted to these systems. Next, we will provide a brief presentation of the methods devised for calculating free energies in quantum systems. Again, it will be shown that many techniques described previously for classical systems, such as PDT, FEP and TI, can be profitably applied when quantum effects are taken into account explicitly. 

In the situation where the transformation involved barrier crossing, e.g., associated with a nonpolar to polar transformation, the computational time was substantially reduced using the X-dynamics formalism, compared with a standard FEP method. This is because X-dynamics searches for alternative lower free energy pathways the coupling parameters (A/ and A2) evolve in the canonical ensemble independently and find a smoother path then when constrained to move as A = A2. Furthermore, a biasing potential in the form... 

The canonical ensemble was developed as the appropriate description of a system in thermal equilibrium with its surroundings by free exchange of energy. Following the discussion of classical systems the density operator of the canonical ensemble is introduced axiomatically as... 

As in the classical expression (25) the quantity ip can be inferred directly as representing the statistical analogue of the Helmholtz free energy. The average behaviour of the canonical ensemble thus obeys the laws of thermodynamics. 

In order to fix ideas, we will consider in this section a canonical ensemble [38], In this ensemble, the Helmholtz free energy is straightforwardly related with the canonical partition function through... 

The YBG equation is a two point boundary value problem requiring the equilibrium liquid and vapor densities which in the canonical ensemble are uniquely defined by the number of atoms, N, volume, V, and temperature, T. If we accept the applicability of macroscopic thermodynamics to droplets of molecular dimensions, then these densities are dependent upon the interfacial contribution to the free energy, through the condition of mechanical stability, and consequently, the droplet size dependence of the surface tension must be obtained. 

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