Statistical mechanics

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Statistical mechanics

Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force.Statistical mechanics, sometimes called statistical physics, can be viewed as a subfield of
physics and
chemistry. Pioneers in establishing the field were Ludwig Boltzmann and Josiah Willard Gibbs.

It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life, therefore explaining thermodynamics as a natural result of statistics and mechanics (classical and quantum) at the microscopic level. In particular, it can be used to calculate the thermodynamic properties of bulk materials from the spectroscopic data of individual molecules.

This ability to make macroscopic predictions based on microscopic properties is the main advantage of statistical mechanics over thermodynamics. Both theories are governed by the second law of thermodynamics through the medium of entropy. However, entropy in thermodynamics can only be known empirically, whereas in statistical mechanics, it is a function of the distribution of the system on its micro-states.

Fundamental postulate

The fundamental postulate in statistical mechanics (also known as the "equal a priori probability postulate") is the following:

:"Given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates."

This postulate is a fundamental assumption in statistical mechanics - it states that a system in equilibrium does not have any preference for any of its available microstates. Given Ω microstates at a particular energy, the probability of finding the system in a particular microstate is "p" = 1/Ω.

This postulate is necessary because it allows one to conclude that for a system at equilibrium, the thermodynamic state (macrostate) which could result from the largest number of microstates is also the most probable macrostate of the system.

The postulate is justified in part, for classical systems, by Liouville's theorem (Hamiltonian), which shows that if the distribution of system points through accessible phase space is uniform at some time, it remains so at later times.

Similar justification for a discrete system is provided by the mechanism of detailed balance.

This allows for the definition of the "information function" (in the context of information theory):...

:$I = - sum_i ho_i ln ho_i = langle ln ho angle.$

When all rhos are equal, I is minimal, which reflects the fact that we have minimal information about the system. When our information is maximal, i.e. one rho is equal to one and the rest to zero (we know what state the system is in), the function is maximal." [Comments: It should be "When all rhos are equal, I is maximal...", please have a check] "This "information function" is the same as the reduced entropic function in thermodynamics.

Microcanonical ensemble

In microcanonical ensemble N, V and E are fixed. Since the second law of thermodynamics applies to isolated systems, the first case investigated will correspond to this case. The "Microcanonical ensemble" describes an isolated system.

The entropy of such a system can only increase, so that the maximum of its entropy corresponds to an equilibrium state for the system.

Because an isolated system keeps a constant energy, the total energy of the system does not fluctuate. Thus, the system can access only those of its micro-states that correspond to a given value "E" of the energy. The internal energy of the system is then strictly equal to its energy.

Let us call $Omega\left(E\right)$ the number of micro-states corresponding to this value of the system's energy. The macroscopic state of maximal entropy for the system is the one in which all micro-states are equally likely to occur, with probability $\left\{1/\left\{Omega \left(E\right)$, during the system's fluctuations.

::$S=-k_Bsum_\left\{i=1\right\}^\left\{Omega \left(E\right)\right\} left \left\{ \left\{1over\left\{Omega \left(E\right) ln\left\{1over\left\{Omega \left(E\right) ight \right\} =k_Bln left\left(Omega \left(E\right) ight\right)$:where :$S$ is the system entropy,:$k_B$ is Boltzmann's constant

Canonical ensemble

In canonical ensemble N, V and T are fixed. Invoking the concept of the canonical ensemble, it is possible to derive the probability $P_i$ that a macroscopic system in thermal equilibrium with its environment, will be in a given microstate with energy $E_i$ according to the Boltzmann distribution:

::

:where ,

The temperature $T$ arises from the fact that the system is in thermal equilibrium with its environment. The probabilities of the various microstates must add to one, and the normalization factor in the denominator is the canonical partition function:

:

where $E_i$ is the energy of the $i$th microstate of the system. The partition function is a measure of the number of states accessible to the system at a given temperature. The article canonical ensemble contains a derivation of Boltzmann's factor and the form of the partition function from first principles.

To sum up, the probability of finding a system at temperature $T$ in a particular state with energy $E_i$ is

:

Thermodynamic Connection

The partition function can be used to find the expected (average) value of any microscopic property of the system, which can then be related to macroscopic variables. For instance, the expected value of the microscopic energy $E$ is "interpreted" as the microscopic definition of the thermodynamic variable internal energy $U$., and can be obtained by taking the derivative of the partition function with respect to the temperature. Indeed,

:

implies, together with the interpretation of $langle E angle$ as $U$, the following microscopic definition of internal energy:

:

The entropy can be calculated by (see Shannon entropy)

:

which implies that

:

is the free energy of the system or in other words,

:

Having microscopic expressions for the basic thermodynamic potentials $U$ (internal energy), $S$ (entropy) and $F$ (free energy) is sufficient to derive expressions for other thermodynamic quantities. The basic strategy is as follows. There may be an intensive or extensive quantity that enters explicitly in the expression for the microscopic energy $E_i$, for instance magnetic field (intensive) or volume (extensive). Then, the conjugate thermodynamic variables are derivatives of the internal energy. The macroscopic magnetization (extensive) is the derivative of $U$ with respect to the (intensive) magnetic field, and the pressure (intensive) is the derivative of $U$ with respect to volume (extensive).

The treatment in this section assumes no exchange of matter (i.e. fixed mass and fixed particle numbers). However, the volume of the system is variable which means the density is also variable.

This probability can be used to find the average value, which corresponds to the macroscopic value, of any property, $J$, that depends on the energetic state of the system by using the formula:

:

where $langle J angle$ is the average value of property $J$. This equation can be applied to the internal energy, $U$:

:

Subsequently, these equations can be combined with known thermodynamic relationships between $U$ and $V$ to arrive at an expression for pressure in terms of only temperature, volume and the partition function. Similar relationships in terms of the partition function can be derived for other thermodynamic properties as shown in the following table; see also the detailed explanation in [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configuration integral] .

References

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*List of notable textbooks in statistical mechanics

* [http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics] article by Lawrence Sklar for the Stanford Encyclopedia of Philosophy.
* [http://www.sklogwiki.org/ Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials.] SklogWiki is particularly orientated towards liquids and soft condensed matter.

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