Entropy microcanonical ensemble

The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

The definition of entropy and the identification of temperature made in the last subsection provides us with a coimection between the microcanonical ensemble and themiodynamics. 

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

The lattice cluster theory (LCT) for glass formation in polymers focuses on the evaluation of the system s configurational entropy Sc T). Following Gibbs-DiMarzio theory [47, 60], Sc is defined in terms of the logarithm of the microcanonical ensemble (fixed N, V, and U) density of states 0( 7),... 

The above realization of the abstract mesoscopic equilibrium thermodynamics is called a Canonical-Ensemble Statistical Mechanics. We shall now briefly present also another realization, called a Microcanonical-Ensemble Statistical Mechanics since it offers a useful physical interpretation of entropy. 

We note that the classical equilibrium entropy (i.e., the eta-function evaluated at equilibrium states) acquires in the context of the Microcanonical Ensemble an interesting physical interpretation. The entropy becomes a logarithm of the volume of the phase space that is available to macroscopic systems having the fixed volume, fixed number of particles and fixed energy. If there is only one microscopic state that corresponds to a given macroscopic state, we can put the available phase space volume equal to one and the entropy becomes thus zero. The one-to-one relation between microscopic and macroscopic thermodynamic equilibrium states is thus realized only at zero temperature. 

Boltzmann6 proposed that at the temperature T = 0, all thermal motion stops (except for zero-point vibration), and the entropy function S can be evaluated by a statistical function W, called the thermodynamic probability W (or, as we will learn in Section 5.2, the partition function Q for a microcanonical ensemble) ... 

Here, S(E) is the entropy in the microcanonical ensemble. Since the density of states of the system is usually unknown, the multicanonical weight factor has to be determined numerically by iterations of short preliminary runs [11,12]. 

The general mathematical formulation of the equilibrium statistical mechanics based on the generalized statistical entropy for the first and second thermodynamic potentials was given. The Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles were investigated as an example. It was shown that the statistical mechanics based on the Tsallis statistical entropy satisfies the requirements of equilibrium thermodynamics in the thermodynamic limit if the entropic index z=l/(q-l) is an extensive variable of state of the system. 

The thermodynamic potential of the microcanonical ensemble, the entropy, is the second thermodynamic potential h =x1=S defined in Eq. (14), which is a function of the variables of state / = , x2=V, x3=N and x4=z. It is obtained from the fundamental thermodynamic potential / by exchanging the variable of state x1 with variable /. In the microcanonical ensemble, the first partial derivatives of the fundamental thermodynamic potential (1) are defined as u1 = T, u2= -p, u3=p, and u4=-E. 

The entropy S for the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble can be written as... 

Substituting Eq. (144) into Eq. (130) for the entropy of the microcanonical ensemble, we obtain the entropy of the canonical ensemble (Eq. (90)). Equation (134) for the pressure of the microcanonical ensemble is identical to Eq. (92) for the pressure of the canonical ensemble. Substituting Eqs. (144) and (86) into Eq. (135) for the chemical potential of the microcanonical ensemble, we obtain Eq. (94) for the chemical potential of the canonical ensemble. Moreover, substituting Eqs. (144) and (86) into Eq. (136) for the variable E of the microcanonical ensemble, we obtain Eq. (96) for the variable E of the canonical ensemble. Thus, for the Tsallis statistics, the canonical and microcanonical ensembles are equivalent in the thermodynamic limit when the entropic parameter z is considered to be an extensive variable of state. 

Section II briefly reviews some arguments about the applicability of one or another ensemble in studying various aspects of small systems. For example, negative heat capacities can be detected in microcanonical ensembles [21-24] if the entropy has a convex dip. The canonical ensemble of the same system does not show any negative heat capacity [25], which is consistent with the general theory for example, the heat capacity is proportional to the energy variance in the canonical ensemble and can never be negative. 

From statistical mechanics the second law as a general statement of the inevitable approach to equilibrium in an isolated system appears next to impossible to obtain. There are so many different kinds of systems one might imagine, and each one needs to be treated differently by an extremely complicated nonequilibrium theory. The final equilibrium relations however involving the entropy are straightforward to obtain. This is not done from the microcanonical ensemble, which is virtually impossible to work with. Instead, the system is placed in thermal equilibrium with a heat bath at temperature T and represented by a canonical ensemble. The presence of the heat bath introduces the property of temperature, which is tricky in a microscopic discipline, and relaxes the restriction that all quantum states the system could be in must have the same energy. Fluctuations in energy become possible when a heat bath is connected to the... 

The simplest situation involves isotropic molecules with no energetic interactions. We may consider an ensemble of systems of equal numbers of component molecules and of energy (microcanonical ensemble). The entropy of mixing involves Boltzmann s formulation of entropy as (24,26)... 

By using the Gibbs entropy S = — p In p for the microcanonical ensemble, the entropy becomes... 

In normal classical statistical mechanics, it is assumed that all states which are fixed by the same external constraints, e.g., total volume V, average energy < ), average particle number N), are equally probable. All possible states of the system are generated and are assigned weight unity if they are consistent with these constraints and, zero otherwise. Thus in the case of an iV-particle system with classical Hamiltonian //j, the microcanonical ensemble entropy S E) is obtained from the total number of states ( ) via the definition... 

Suppose (9,11) holds at Iq- Then, for instance, using a microcanonical ensemble, where A = —TS, (i.e., only configurational entropies are used)... 

On the microcanonical ensemble, entropy is a measure of the total available phase. (See the Further Reading section at the end of this chapter.)... 

The proper thermodynamic potential for the microcanonical ensemble is the entropy ... 

When the constraints are (T, V, p), energy and particles can exchange across the boundary. This is called the Grand Canonical ensemble. It is important for processes of ligand binding, and is illustrated in Chapter 28. When (U,V,N) are held constant at the boundary, no extensive quantity exchanges across the boundary, so there is no bath, and maximum entropy 5([/, V,N) identifies the state of equilibrium (see Chapter 7). This is called the Microcanonical ensemble. 

We begin with the classical ideal gas, in which any quantum mechanical features of the constituent particles are neglected. This is actually one of the few systems that can be treated in the microcanonical ensemble. In order to obtain the thermodynamics of the classical ideal gas in the microcanonical ensemble we need to calculate the entropy from the total number of states a(E, S2) with energy E and volume S2, according to Eq. (D.28). The quantity a E, S2) is given by... 

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