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The ensemble formalises the notion that a physicist can imagine repeating an experiment again and again under the same macroscopic conditions, but, unable to control the microscopic details, may expect to observe a range of different outcomes. When an ensemble has an infinite number of members, it can be seen as defining a Probability Measure on the State Space ( Phase Space ) of the system. Even though the dynamics of the real single system (for example, a complete gas of molecules, or a complete stockmarket) may be uncalculably complex, or Stochastic , or even Discontinuous , the ''average'' (statistical) properties of the ensemble of possibilities as a whole may remain well defined, smoothly evolving, or for systems at macroscopic Equilibrium even Stationary . The word ensemble is also sometimes used for smaller sets of possibilities, Sampled from the full set of possible states. Thus for example, an ensemble of walkers in a Markov Chain Monte Carlo iteration; or an Ensemble Forecast in Meteorology , where a whole ensemble of possible initial states is projected forwards, to try to give an idea of the spread of possible forecast outcomes; or Climate Ensembles , where the space of macroscopically possible perturbations of the model physics is also sampled. The notional size of the mental ensembles in thermodynamics, statistical mechanics and Quantum Statistical Mechanics can be very large indeed, to include every possible Microscopic State the system could be in, consistent with its observed Macroscopic properties. But for important physical cases it can be possible, by clever mathematical manipulations, to calculate averages directly over the whole of the thermodynamic ensemble, to obtain explicit formulas for many of the thermodynamic quantities of interest, often in terms of the Partition Function ''Z'', which encodes the underlying physical structure of the system. Some of these results are presented in the article Statistical Mechanics . If the system states are ''perfectly mixing'' ( Ergodic ), and the ensemble is large and corresponds to a probability measure which is ''invariant'' under this dynamics, then the time-average of a quantity taken over a sufficiently long time for a single real evolving system ''should'' be well predicted by the Ensemble Average , averaged over the members of the ensemble as a whole -- ie the average value for an instant observation notionally repeated in a lot of different systems. This is known as the Ergodic Hypothesis . If not, one may infer that there is more macroscopically discoverable information available than one first thought about the microscopic state of the system, which may be usable to create a better- Conditioned ensemble. The word ''ensemble'' is particularly used in thermodynamics; by some physicists working in Bayesian Probability theory; and by mathematicians whose work in probability theory is heavily influenced by physicists, especially those working on Random Matrices . Most "pure" mathematicians working in Probability Theory do not use the term, preferring to use the terminology of Probability Spaces . PRINCIPAL ENSEMBLES OF STATISTICAL THERMODYNAMICS Different macroscopic environmental constraints lead to different types of ensembles, with particular statistical characteristics. The following are the most important:
The calculations which can be made over each of these ensembles are explored further in the article Statistical Mechanics . The main result for each ensemble however, is its characteristic state function: Microcanonical: Canonical: Grand canonical: Other thermodynamic ensembles can be also defined, corresponding to different physical requirements, for which analogous formulae can often similarly be derived. PROPERTIES OF "GOOD" ENSEMBLES
pdf over the states in the ensemble should reflect their equilibrium pdf. Gibbs State .
Time-averages of macroscopic quantities of interest will only have a chance of being the same as ensemble-averages, if the system evolving over time can actively explore close to all of the State Space possibilities included in the ensemble (ergodicity). Otherwise the probability density over the ensemble will not be representative of the probability density of states in the time-evolution. (See Ergodic Hypothesis ). ENSEMBLES IN QUANTUM STATISTICAL MECHANICS ''See main article: Quantum Statistical Mechanics '' Putting aside for the moment the question of how statistical ensembles are generated Operationally , we should be able to perform the following two operations on ensembles ''A'', ''B'' of the same system:
Under certain conditions therefore, Equivalence Class es of statistical ensembles have the structure of a convex set. In quantum physics, a general Model for this convex set is the set of Density Operators on a Hilbert Space . Accordingly, there are two types of ensembles: |
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