Thermodynamic averaging

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. 

Thermodynamic average. An average over all points in phase space at a single time. 

To handle broken ergodicity in the calculation of thermodynamic averages (i.e., time-independent averages of the type (A) oc / /I (x) exp(— ft77(x))dx) a host of methods have been devised [2]. In a subset of these, the Boltzmann distribution is altered and replaced with a more delocalized one, w(E). One is thus able to generate (faster) samples distributed according to w and subsequently, one can use importance... 

Here an additional distinction is to be made between thermodynamic averages of a conformational observable such as the internal energy, which converges well if potential minima are correctly sampled, and statistical properties such as free energies, which depend on the entire partition function. 

In a recent development, Corcelli et al. [110] introduced a convenient bias function with general applicability that promises to accelerate the convergence of rate calculations in systems with large enthalpy barriers. They apply a puddle potential (used previously by the same group to enhance thermodynamic averaging [69]) that changes the potential energy surface from which the trajectories are initiated to become... 

Andricioaei, I. Dinner, A.R. Karplus, M., Self-guided enhanced sampling methods for thermodynamic averages, 7. Chem. Phys. 2003,118, 1074—1084... 

The use of energy minimization can be extended to solid solutions and highly non-stoichiometric compounds. In principle the method is simple we take a suitable thermodynamic average over the results of minimizations of different possible arrangements of the atoms. The overall procedure for a solution A0.5B0.5 is then as follows ... 

Determine the thermodynamic properties of the system by taking a thermodynamic average. For example, the enthalpy is given by... 

Continuous transitions are characterized by a critical behavior of a physical observable, which is the order parameter y of the transition. Above the transition temperature the thermodynamic average of the order parameter is zero, indicative of complete loss of long-range order. Below Tc the value of is nonzero indicating long-range order, and follows a power law when approaching the critical temperature ... 

The thermodynamical average 5 (a) over the Legendre polynomials P occur in the expressions for the susceptibilities, the specific heat, and the dipolar fields in Section II.B. For uniaxial anisotropy these averages read... 

Fine-Structure Measurements of Oxygen A Band Absorbance for Estimating the Thermodynamic Average Temperature of the Earth s Atmosphere 136... 

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