Microcanonical ensemble, definition

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 the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index z is an extensive variable of state of the system. In this case the thermodynamic quantities of the Tsallis statistics belong to one of the classes of homogeneous functions of the first or zero orders. 

By definition, the microcanonical ensemble contains all possible configurations in the 6N-dimensionaI phase space with the same energy and a constant probability of being in each configuration N is the number of particles in the system under consideration. This ensemble describes an isolated system with constant N and V, or constant N and zero external pressure [28]. Constant-energy simulations are not recommended for equilibration because, without the energy flow facilitated by the temperature control methods, the desired temperature cannot be achieved. However, during the data collection phase, if one is... 

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... 

When the Boltzmann distribution was derived in Section 5.2, we assumed a constant number of particles (N) and a given volume (V). We also assumed that the energy was constant. In the canonical ensanble, we imagine that the subsystems (members of the canonical ensemble) may exchange energy but not particles with each other. The ensemble that we are supposed to use is the one where the assumptions agree with the experimental conditions. Since the total energy is constant in the microcanonical ensemble, the definition of such a simple concept as temperature has to be done indirectly. 

Redo exercise 17.13, except for a microcanonical ensemble. The definition of the microcanonical ensemble is in the text. 

We introduced the microcanonical analysis in Section 2.7 and found that the density of states g E) already contains all relevant information about the phases of the system. Alternatively, one can also use the phase space volume AG(E) of the energetic shell that represents the macrostate in the microcanonical ensemble in the energetic interval (E,E+ AE) with AE being sufficiently small to satisfy AG E)=g E)AE. In the limit AE —> 0, the total phase space volume up to the energy E can thus be expressed as G E) = dE g(E ). Since g E) is positive for all E, G(E) is a monotonically increasing function and this quantity is suitably related to the microcanonical entropy S(E) of the system. In the definition of Hertz,... 

An ensemble is defined by the density of states p included in it. In principle all states that satisfy the imposed external conditions are considered to be members of the ensemble. There are three types of ensembles the microcanonical, canonical and grand canonical. The precise definition of the density p for each ensemble depends on the classical or quantum mechanical nature of the system. 

---