Free energy quenched averaged

For critical quench experiments there is a synnnetry < )q = 0 and from equation (A3,3,50) S( ) = ( ), leading to a syimnetric local free energy ( figure A3,3,6) and a scaled order parameter whose average is zero, 8v / = i /. For off-critical quenches this synnnetry is lost. One has 8( ) = ( ) -t ( ) which scales to 5 j/ = with... 

Since tire charge variables are quenched tire tliennodynamics of tire system requires averaging tire free energy using tire distribution P(X), i.e. 

Since the charge variables X. are quenched the thermodynamics of the system requires averaging the free energy using the distribution P(kX i.e. 

We will consider the case of an isolated SAW, of length n. Two types of averaging are usually considered, corresponding to the quenched and annealed cases respectively. In the quenched case, the average is taken of the logarithm of the partition function over the entire distribution, so that the free energy is... 

The averages are taken over all a. It is noteworthy that the annealed free energy is an upper bound on the quenched free energy. This is useful as the former is usually easier to calculate. Better approximations are often provided by the Morita [15] approximation and the rephca method. We refer readers to [2] and references therein for further details. 

As far we consider the case of quenched disorder, the free energy of the system is obtained by averaging the logarithm of the partition function Z over the disorder distribution [15] this amounts to use so-called replica trick [52] writing the logarithm in the... 

Applying the replica method in order to average the free energy over different configurations of quenched disorder one finds the effective Hamiltonian of the m-vector model with long-range-correlated disorder [65] ... 

To perform the average of the free energy over the quenched disorder, the replica method is applied and for the n-replicated partition function we obtain ... 

In order to compute quenched averages of the polymer chain we will have to solve the n—body replicated partition sum given in Eq. (40). This path integral cannot be evaluated analytically and a variational approach has been used in Refs. [22,8] to make further progress. The procedure is to follow the work of Feynman [7] and others [23,24] and model Hn by a solvable trial Hamiltonian /i which is determined by the stationarity of the variational free energy... 

To discuss this new type of ordered phase in spin glasses, one would like to have a microscopic model where the actual interactions and anisotropies are considered and the average over a realistic description of the site dilution disorder is performed. Clearly this is a difficult task and up to now no realistic model of a spin glass has been solved analytically. In addition, there exists another difficulty because a proper treatment of systems with quenched disorder like spin glasses involves averaging the free energy F rather than the partition function Z... 

This section is devoted to the proof of the existence of the quenched free energy. The free energy of a disordered system is a priori a random variable, however we will show that it is a degenerate random variable, i.e. a constant this phenomenon is called self-averaging property of the free energy. 

The first step in the proof is controlling the so-caUed quenched averaged free energy by the bound on Hn,u(t) in point (1) in Section 4.1, and by the hypothesis on cu, logZ L. As a matter of fact the same bound implies directly... 

So that the sequence with which we wish to define, in the limit, the quenched averaged free energy is bounded above. By using the same esti-... 

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