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## Statistical mechanics
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 asset 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 postulateThe fundamental postulate in statistical mechanics (also known as the *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 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 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. This "information function" is the same as the ## Microcanonical ensembleSince the second law of thermodynamics applies to isolated systems, the first case investigated will correspond to this case. The 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 Let us call 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 during the system's fluctuations. - where
- is the system entropy,
- is Boltzmann's constant
## Canonical ensembleInvoking the concept of the canonical ensemble, it is possible to derive the probability that a macroscopic system in thermal equilibrium with its environment will be in a given microstate with energy : - where ,
The temperature 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 is the energy of the 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 in a particular state with energy is ## Thermodynamic ConnectionThe 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 is implies, together with the interpretation of as , 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 (internal energy), (entropy) and (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 , 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 with respect to the (intensive) magnetic field, and the pressure (intensive) is the derivative of 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, where is the average value of property . This equation can be applied to the internal energy, : Subsequently, these equations can be combined with known thermodynamic relationships between and 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 configuration integral.
To clarify, this is not a grand canonical ensemble. It is often useful to consider the energy of a given molecule to be distributed among a number of modes. For example, translational energy refers to that portion of energy associated with the motion of the center of mass of the molecule. Configurational energy refers to that portion of energy associated with the various attractive and repulsive forces between molecules in a system. The other modes are all considered to be internal to each molecule. They include rotational, vibrational, electronic and nuclear modes. If we assume that each mode is independent (a questionable assumption) the total energy can be expressed as the sum of each of the components: Where the subscripts , , , , , and correspond to translational, configurational, nuclear, electronic, rotational and vibrational modes, respectively. The relationship in this equation can be substituted into the very first equation to give:
Thus a partition function can be defined for each mode. Simple expressions have been derived relating each of the various modes to various measurable molecular properties, such as the characteristic rotational or vibrational frequencies. Expressions for the various molecular partition functions are shown in the following table.
These equations can be combined with those in the first table to determine the contribution of a particular energy mode to a thermodynamic property. For example the "rotational pressure" could be determined in this manner. The total pressure could be found by summing the pressure contributions from all of the individual modes, ie: ## Grand canonical ensembleIf the system under study is an open system, (matter can be exchanged), where N Let's rework everything using a grand canonical ensemble this time. The volume is left fixed and does not figure in at all in this treatment. As before,
## Equivalence between descriptions at the thermodynamic limitAll the above descriptions differ in the way they allow the given system to fluctuate between its configurations. In the micro-canonical ensemble, the system exchanges no energy with the outside world, and is therefore not subject to energy fluctuations, while in the canonical ensemble, the system is free to exchange energy with the outside in the form of heat. In the thermodynamic limit, which is the limit of large systems, fluctuations become negligible, so that all these descriptions converge to the same description. In other words, the macroscopic behavior of a system does not depend on the particular ensemble used for its description. Given these considerations, the best ensemble to choose for the calculation of the properties of a macroscopic system is that ensemble which allows the result be most easily derived. ## Random walkersThe study of long chain polymers has been a source of problems within the realms of statistical mechanics since about the 1950's. One of the reasons however that scientists were interested in their study is that the equations governing the behaviour of a polymer chain were independent of the chain chemistry. What is more, the governing equation turns out to be a random (diffusive) walk in space. Indeed, Schrodinger's equation is itself a diffusion equation in imaginary time, ## Random walks in timeThe first example of a random walk is one in space, whereby a particle undergoes a random motion due to external forces in its surrounding medium. A typical example would be a pollen grain in a beaker of water. If one could somehow "dye" the path the pollen grain has taken, the path observed is defined as a random walk. Consider a toy problem, of a train moving along a 1D track in the x-direction. Suppose that the train moves either a distance of + or - a fixed distance ; due to The second quantity is known as the correlation function. The delta is the kronecker delta which tells us that if the indices
As stated is 0, so the sum of 0 is still 0. It can also be shown, using the same method demonstrated above, to calculate the root mean square value of problem. The result of this calculation is given below
From the diffusion equation it can be shown that the distance a diffusing particle moves in a media is proportional to the root of the time the system has been diffusing for, where the proportionality constant is the root of the diffusion constant. The above relation, although cosmetically different reveals similar physics, where ## Random walks in spaceRandom walks in space can be thought of as snapshots of the path taken by a random walker in time. One such example is the spatial configuration of long chain polymers. There are two types of random walk in space: By considering a freely jointed, non-interacting polymer chain, the end-to-end vector is where is the vector position of the Assuming, as stated, that that distribution of end-to-end vectors for a very large number of identical polymer chains is gaussian, the probability distribution has the following form
What use is this to us? Recall that according to the principle of equally likely
where c is an arbitrary proportionality constant. Given our distribution function, there is a maxima corresponding to . Physically this amounts to there being more microstates which have an end-to-end vector of 0 than any other microstate. Now by considering ## See also- Fluctuation dissipation theorem
- Important Publications in Statistical Mechanics
- List of notable textbooks in statistical mechanics
- Ising Model
- Mean field theory
- Ludwig Boltzmann
- Paul Ehrenfest
- Thermodynamic limit
- Nanomechanics
- Statistical physics
## References- Chandler, David (1987).
*Introduction to Modern Statistical Mechanics*. Oxford University Press.__ISBN 0-19-504277-8__. - Huang, Kerson (1990).
*Statistical Mechanics*. Wiley, John & Sons, Inc.__ISBN 0-471-81518-7__. - Kroemer, Herbert; Kittel, Charles (1980).
*Thermal Physics (2nd ed.)*. W. H. Freeman Company.__ISBN 0-7167-1088-9__. - McQuarrie, Donald (2000).
*Statistical Mechanics (2nd rev. ed.)*. University Science Books.__ISBN 1-891389-15-7__. - Dill, Ken; Bromberg, Sarina (2003).
*Molecular Driving Forces*. Garland Science.__ISBN 0-8153-2051-5__. - List of notable textbooks in statistical mechanics
Categories: Statistical mechanics | Thermodynamics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Statistical_mechanics". A list of authors is available in Wikipedia. |