Temperatures: old, new and middle aged

Abstract

Abstract This study explores the various ways in which a 'temperature' can be introduced and calculated in a molecular simulation. We focus on the relatively recent formula of Rugh, and a simplified version called the 'configurational' temperature, T config, which became popular after Rugh's work. We derive formulae for these various 'temperatures' for the special case of the soft–sphere fluid, with the pair interaction, , where ε and σ set the energy and length scales, respectively, and for different n values. We show with a number of simple test cases that the various prescriptions of temperature give the same result as the thermodynamic temperature at equilibrium. Although much of the current work uses the configurational temperature, Rugh's temperature, T R, has the advantage that it has the thermodynamic value in circumstances where the configurational temperature is not defined (such as in the limit of the ideal gas, where Rugh's temperature becomes the kinetic temperature). The configurational temperature is a particular example of the hypervirial theorem. Using the particle forces and their gradients, we calculate the first and second hypervirial temperatures, which we find to be equal, even though the force moments (i.e. ) and their derivatives themselves do not conform to a Gaussian distribution. Various moment ratios were shown to have a linear dependence on the reciprocal of the density, at not too high densities. We explored the behaviour of the kinetic and configurational temperatures for some non–equilibrium systems, where quite different trends were observed in these two quantities in any transient or non–stationary stage. The kinetic temperature was found to respond more rapidly to perturbations than the configurational temperature, as might be expected since usually energy is more rapidly transported than matter. Acknowledgements DMH would like to thank the Engineering and Physical Sciences Research Council of Great Britain (EPSRC) for funding workstations on which the calculations presented in this work were performed.