Ensemble Laplace transformation

Pearson, E. M. Halicioglu, T. Tiller, W. A., Laplace-transform technique for deriving thermodynamic equations from the classical microcanonical ensemble, Phys. Rev. A 1988, 32, 3030-3039... 

If Q (E) is differentiable in the ordinary sense the partition function of a generalized ensemble with m intensive parameters is the m-fold Laplace transform of the microcanonical partition function e, ... 

By taking the Laplace transform of eqn. (290) and then multiplying throughout by the initial velocity u(0) and finally taking an ensemble average, the Laplace transformed velocity autocorrelation function is [490]... 

As we showed in Section 2.5.4, thermal averages in isostress isostrain ensembles can be related through a Laplace transformation. Hence, for the conjugate stress r and strain A, wc may employ Echange variables according to Tja — t and —> A giving... 

The quantities qi, qi and have the dimensions cm-1. They are inverse screening lengths. Later we shall see that qo also plays a role in the equilibrium structure of the ionic solution. Equation (9.2.15) can be solved by means of Laplace transformation with respect to time. When the solutions are multiplied by <5c (q, 0) and Sc (q, 0) and ensemble-averaged we obtain... 

Because the equilibrium ensemble is time translationally invariant, C is a function of t — t. Since we will be more interested in the Fourier and Laplace transforms of (19), we introduce the definitions... 

However, the principles and the techniques of renormalization theory are not directly related to the existence of fields. They apply whenever one deals with a critical system, i.e. whenever one has to describe large-scale phenomena which depend only globally on the chemical microstructure. Thus, because an ensemble of long polymers in a solution constitutes a critical system, renormalization principles and renormalization techniques must be directly applicable to their study. Actually, this idea appeared quite naturally. It led to the decimation method which has been described previously and which lacks efficiency. However, the same idea can be applied in a much better way. This direct renormalization method (des Cloizeaux 1980)37,38 consists in adapting to polymers methods which had been successful in field theory.39 In other words, the aim is to bypass the Laplace de Gennes transformation (see Chapter 11). This method applies to semi-dilute solutions as well as to dilute solutions. 

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