Entropy 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) ... 

For equilibrium systems, diemiodynamic entropy is related to ensemble density distribution p as... 

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. 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

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 Gibbs free energy is given in terms of the enthalpy and entropy, G — H — TS. The enthalpy and entropy for a macroscopic ensemble of particles may be calculated from properties of the individual molecules by means of statistical mechanics. 

Since entropy is maximum when the system is maximally disordered, Smeas i) takes its maximal value of one binary bit per site for an equiprobable ensemble Sset t oo) = 1 only if all states are cyclic. If the system settles onto a unique cyclic... 

For reven sible systems, evolution almost always leads to an increase in entropy. The evolution of irreversible systems, one the other hand, typically results in a decrease in entropy. Figures 3.26 and 3.27 show the time evolution of the average entropy for elementary rules R32 (class cl) and R122 (class c3) for an ensemble of size = 10 CA starting with an equiprobable ensemble. We see that the entropy decreases with time in both cases, reaching a steady-state value after a transient period. This dc crease is a direct reflection of the irreversibility of the given rules,... 

The most important characteristic of self information is that it is a discrete random variable that is, it is a real valued function of a symbol in a discrete ensemble. As a result, it has a distribution function, an average, a variance, and in fact moments of all orders. The average value of self information has such a fundamental importance in information theory that it is given a special symbol, H, and the name entropy. Thus... 

As an example of self information and entropy, consider the ensemble, U, consisting of the binary digits, 0, and 1. Then if we letp = Pr(l),... 

In a channel context, with X as the input ensemble, and Y as the output ensemble, we can interpret H(X Y) as the average additional information required at the outpiit to specify an input when the output is given thus H(X Y) is known as equivocation. Similarly, I/(F. Z) can be interpreted as the part of the entropy of Y that is not information about X, and thus H( y X) is known as noise. 

Thus the entropy of a product ensemble, X Y, can only be reduced by statistical dependence between the X and Y ensembles. 

Problem—Show that the entropy of an ensemble is a convex upward function of the probabilities of the points in the ensemble. 

There is thus assumed to be a one-to-one correspondence between the most probable distribution and the thermodynamic state. The equilibrium ensemble corresponding to any given thermodynamic state is then used to compute averages over the ensemble of other (not necessarily thermodynamic) properties of the systems represented in the ensemble. The first step in developing this theory is thus a suitable definition of the probability of a distribution in a collection of systems. In classical statistics we are familiar with the fact that the logarithm of the probability of a distribution w[n is — J(n) w n) In w n, and that the classical expression for entropy in the ensemble is20... 

Entropy and Equilibrium Ensembles.—If one can form an algebraic function of a linear operator L by means of a series of powers of L, then the eigenvalues of the operator so formed are the same algebraic function of the eigenvalues of L. Thus let us consider the operator IP, i.e., the statistical matrix, whose eigenvalues axe w ... 

With this definition, the classical entropy per system equals the ensemble average of the expectation value of 8 in occupation number representation. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

According to the latter model, the crystal is described as formed of anumber of equal scatterers, all randomly, identically and independently distributed. This simplified picture and the interpretation of the electron density as a probability distribution to generate a statistical ensemble of structures lead to the selection of the map having maximum relative entropy with respect to some prior-prejudice distribution m(x) [27, 28],... 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

---