Statistical mechanics is the study of large groups of particles in equilibrium. While the individual behavior of each particle can be described with classical or quantum mechanics, this becomes unfeasible with large numbers of particles. Statistical mechanics deals with observable behaviors that can be deduced from the nature of the different particles by using probability. It is the link from the world of single photons and electrons to the world of our everyday lives.
Some elements of statistical mechanics are useful for any type of system. The concept of equipartition of energy states that for each degree of freedom, the system will have energy proportional to 1/2kT, where k is Boltzmanns' constant and T is the temperature. That is, for each way in which the system can be changed in position or momentum, the energy will increase according to the above rule. The equipartition of energy is useful for estimating the temperature-related behavior of different substances, an application of thermodynamics that crosses the line into statistical mechanics. Ideas such as enthalpy, entropy, pressure, and temperature, are also useful in both fields.
The basic part of statistical mechanics is the equation of state. This equation links general system properties like volume, pressure, temperature, and number of particles. (The most famous equation of state is the ideal gas law: PV = nRT, where P is pressure, V is volume, n is the number of particles, R is a constant, and T is temperature.) There are three different ways to arrive at an equation of state, through what are called different ensembles: the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble. Each makes a different but equivalent mathematical assumption about equilibrium states, allowing the equation of state to be derived in three equivalent ways.
The microcanonical ensemble assumes that the particles in the system have a total energy that is known. Each set of particle energy states that would sum to give that total energy is equally likely. The microcanonical ensemble is good for dealing with systems that are isolated from the rest of the universe, so that no transfer of energy can occur.
The canonical ensemble is not in isolation but has a fixed number of particles. The system's contact with the outside world is limited to transfer of energy into the particles that are already inside the system. This ensemble uses in its derivation the fact that temperatures are constant at equilibrium. The temperature, not the total energy, is assumed to be known.
Finally, the grand canonical ensemble does not require a fixed number of particles, merely a known average number of particles. It, like the canonical ensemble, is in contact with the outside world. The property which is assumed to be constant is the chemical potential, defined as the change in free energy when the number of particles changes but the temperature and volume are known and constant.
The ensembles describe differences in the particle's state: whether it's isolated and what properties are constants. There are also differences in the properties of the particles themselves, and those differences change the behavior significantly. There are three basic ways of expressing particle properties in statistical mechanics situations, and they are reflected in the distributions of particles in different states. They are the Maxwell-Boltzmann distribution, the Bose-Einstein distribution, and the Fermi-Dirac distribution.
The Maxwell-Boltzmann distribution describes essentially classical particles. It is assumed that most possible states of this system are not filled. This distribution is most commonly used for gases but will work for any equilibrium system with gas-like conditions. Low densities and high temperatures will make these statistics work most like a classical gas and will minimize interactions and quantum mechanical behavior.
The Fermi-Dirac distribution applies to quantum mechanical systems of fermions, particles that obey the Pauli exclusion principle. The largest number of particles possible in a given state is one. Therefore, the statistics derived by Enrico Fermi and Paul Dirac can be used to describe the behavior of equilibrium systems with this behavior, such as electron behavior in metals. This type of statistical mechanics can be used to describe the behavior of electrons in metals as it pertains to temperature, as well as having other applications.
The Bose-Einstein distribution applies to quantum mechanical systems of bosons. Since there is no restriction on the number of particles that can be in the same state here, at the lowest temperatures, all particles may be forced into the same ground state. This is known as a Bose-Einstein condensate.
Using these different ensembles and distributions, average behavior can be calculated or predicted for a wide variety of physical systems in equilibrium, and many thermodynamic conclusions can be verified. Statistical mechanics offers insight into such diverse physical systems as a balloon, a Bose-Einstein condensate, and a neutron star. The applications of statistical mechanics in the rest of physics are nearly limitless.
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