Thennodynamic limit

If this condition is not satisfied, there is no unique way of calculating the observed value of ff, and the validity of the statistical mechanics should be questioned. In all physical examples, the mean square fluctuations are of the order of 1/Wand vanish in the thennodynamic limit. 

This result is identical to that obtained from a canonical ensemble approach in the thennodynamic limit, where the fluctuations in N vanish and (N) = N. The single-particle expression for the canonical partition fiinction = (-" can be evaluated using ih r rV i f<,2M) or a particle in a cubical box of volume V. 

In the thennodynamic limit (N x, F -> oo withA7F= p), this is just the virial expansion for the pressure, with 7,(7) identified as the second virial coefficient... 

Both systems give the same results in the thennodynamic limit. We discuss the solution for the open chain at zero field and the closed chain for the more general case of H 0. 

The free energy G in the thennodynamic limit (N oo) follows from... 

It is important to recognize that thennodynamic laws are generalizations of experimental observations on systems of macroscopic size for such bulk systems the equations are exact (at least within the limits of the best experimental precision). The validity and applicability of the relations are independent of the correchiess of any model of molecular behaviour adduced to explain them. Moreover, the usefiilness of thennodynamic relations depends cmcially on measurability, unless an experimenter can keep the constraints on a system and its surroundings under control, the measurements may be worthless. 

As we shall see, because of the limitations that the second law of thennodynamics imposes, it may be impossible to find any adiabatic paths from a particular state A to another state B because In this... 

In the Lewis and Gibson statement of the third law, the notion of a perfect crystalline substance , while understandable, strays far from the macroscopic logic of classical thennodynamics and some scientists have been reluctant to place this statement in the same category as the first and second laws of thennodynamics. Fowler and Guggenheim (1939), noting drat the first and second laws both state universal limitations on processes that are experunentally possible, have pointed out that the principle of the unattainability of absolute zero, first enunciated by Nemst (1912) expresses a similar universal limitation ... 

The principle of tire unattainability of absolute zero in no way limits one s ingenuity in trying to obtain lower and lower thennodynamic temperatures. The third law, in its statistical interpretation, essentially asserts that the ground quantum level of a system is ultimately non-degenerate, that some energy difference As must exist between states, so that at equilibrium at 0 K the system is certainly in that non-degenerate ground state with zero entropy. However, the As may be very small and temperatures of the order of As/Zr (where k is the Boltzmaim constant, the gas constant per molecule) may be obtainable. 

This behaviour is characteristic of thennodynamic fluctuations. This behaviour also implies the equivalence of various ensembles in the thermodynamic limit. Specifically, as A —> oo tire energy fluctuations vanish, the partition of energy between the system and the reservoir becomes uniquely defined and the thennodynamic properties m microcanonical and canonical ensembles become identical. 

If z = exp(pp) l, one can also consider the leading order quantum correction to the classical limit. For this consider tlie thennodynamic potential cOq given in equation (A2.2.144). Using equation (A2.2.149). one can convert the sum to an integral, integrate by parts the resulting integral and obtain the result ... 

If 5 aj(r) is slowly varying in space, tire long-wavelength limit x-j(k —> 0) reduces to a set of static susceptibilities or thennodynamic derivatives. Now, since for t > 0 the external fields are zero, it is useful to evaluate the one-sided transfomi... 

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

In section C2.5.2 we considered a variational-type theory to treat the thennodynamics of the random hydrophobic-hydrophilic heteropolymer. Here we describe a limiting behaviour of the random bond model [30]. 

Templates nonnally bind too su ongly to their products to operate catalytically product inhibition severely limits turnover. If a template bound tlie product less strongly than either tlie transition state or the starting material, it would be less susceptible to product inhibition. Such catalytic templates have an intellectual appeal as enzyme mimics as well as practical advantages. In nearly all cases published so far there are insufficient kinetic and thennodynamic data available to assess whether the transition state is bound more or less strongly tlian the product. It will be fascinating to see whether it is... 

As an example, thennodynamic prediction of existence conditions for YBa2Cus09 gives a stability limit of 470°C in ambient oxygen (Voronin 1994a). Also a structural combination between the 124 and 123 phases is possible ... 

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