NationMaster - Encyclopedia: 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. The microcanonical ensemble is the simplest of the ensembles of statistical mechanics. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ... The isothermalâ€“isobaric ensemble is a statistical mechanical ensemble where the system is allowed to exchange energy with a heat bath of temperature T and the volume can also change though its mean value is tuned by the pressure P applied. ... It has been suggested that this article or section be merged with Probability axioms. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ...

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. Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...

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. Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...

1 Fundamental postulate
2 Microcanonical ensemble
3 Canonical ensemble
- 3.1 Thermodynamic Connection
4 Grand canonical ensemble
5 Equivalence between descriptions at the thermodynamic limit
6 Random walkers
- 6.1 Random walks in time
- 6.2 Random walks in space
7 See also
8 References
9 External links

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. In statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system, that the system visits in the course of its thermal fluctuations. ...

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. In mathematical physics, Liouvilles theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. ... For other senses of this term, see phase space (disambiguation). ...

Similar justification for a discrete system is provided by the mechanism of detailed balance. In mathematics, and in statistical mechanics in physics, a Markov process is said to show detailed balance if the transition rates between each pair of states i and j in the state space obey where P is the Markov transition matrix, ie Pij = P( Xt =i | Xtâˆ’1 = j ); and...

This allows for the definition of the information function (in the context of information theory): A bundle of optical fiber. ...

$I = sum_i rho_i lnrho_i = langle ln rho rangle$

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 reduced entropic function in thermodynamics.

Microcanonical ensemble

Main article: Microcanonical ensemble

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 microcanonical ensemble is the simplest of the ensembles of statistical mechanics. ... The second law of thermodynamics is an expression of the universal law of increasing entropy. ... Isolation can refer to: Isolation as a psychological phenomenon. ... Isolation can refer to: Isolation as a psychological phenomenon. ...

The entropy of such a system can only increase, so that the maximum of its entropy corresponds to an equilibrium state for the system. Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ...

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. In thermodynamics, an isolated system, as contrasted with a closed system, is a physical system that does not interact with its surroundings. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

Let us call $Omega(E)$ 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. Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ...

$S=k_Bln left(Omega (E) right)$

where

$S$ is the system entropy,

$k_B$ is Boltzmann's constant

Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

Canonical ensemble

Main article: Canonical ensemble

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 $E_i$ : A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ...

$P_i = {expleft(-beta E_iright)over{sum_j^{j_{max}}expleft(-beta E_jright)}}$

where $beta={1over{kT}}$ ,

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: Broadly, normalization (also spelled normalisation) is any process that makes something more normal, which typically means conforming to some regularity or rule, or returning from some state of abnormality. ... In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...

$Z = sum_j^{j_{max}} expleft(-beta E_jright)$

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. A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ...

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

$P_i = frac{exp(-beta E_i)}{Z}$

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,

$langle Erangle={sum_i E_i e^{-beta E_i}over Z}=-{1 over Z} {dZ over dbeta}$

implies, together with the interpretation of $<E>$ as $U$ , the following microscopic definition of internal energy: In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

$Ucolon = -{dln Zover d beta}.$

The entropy can be calculated by (see Shannon entropy) Entropy of a Bernoulli trial as a function of success probability. ...

${Sover k} = - sum_i p_i ln p_i = sum_i {e^{-beta E_i}over Z}(beta E_i+ln Z) = ln Z + beta U$

which implies that

$-frac{ln(Z)}{beta} = U - TS = F$

is the Free energy of the system or in other words, The free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ...

$Z=e^{-beta F},$

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). In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... The free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ...

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:

$langle J rangle = sum_i p_i J_i = sum_i J_i frac{exp(-beta E_i)}{Z}$

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

$U = sum_i E_i frac{exp(-beta E_i)}{Z}$

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.

Helmholtz free energy:	$F = - {ln Zover beta}$
Internal energy:	$U = -left( frac{partialln Z}{partialbeta} right)_{N,V}$
Pressure:	$P = -left({partial Fover partial V}right)_{N,T}= {1over beta} left( frac{partial ln Z}{partial V} right)_{N,T}$
Entropy:	$S = k (ln Z + beta U),$
Gibbs free energy:	$G = F+PV=-{ln Zover beta} + {Vover beta} left( frac{partial ln Z}{partial V}right)_{N,T}$
Enthalpy:	$H = U + PV,$
Constant Volume Heat capacity:	$C_V = left( frac{partial U}{partial T} right)_{N,V}$
Constant Pressure Heat capacity:	$C_P = left( frac{partial H}{partial T} right)_{N,P}$
Chemical potential:	$mu_i = -{1over beta} left( frac{partial ln Z}{partial N_i} right)_{T,V,N}$

To clarify, this is not a grand canonical ensemble. Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist. ... The free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... Josiah Willard Gibbs (February 11, 1839 â€“ April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... The free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or Î”H, or rarely as Ï‡) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a... In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ...

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:

$E = E_t + E_c + E_n + E_e + E_r + E_v,$

Where the subscripts $t$ , $c$ , $n$ , $e$ , $r$ , and $v$ 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:

$Z = sum_i expleft(-beta(E_{ti} + E_{ci} + E_{ni} + E_{ei} + E_{ri} + E_{vi})right)$

$= sum_i expleft(-beta E_{ti}right) expleft(-beta E_{ci}right) expleft(-beta E_{ni}right) expleft(-beta E_{ei}right) expleft(-beta E_{ri}right) expleft(-beta E_{vi}right)$

If we can assume all these modes are completely uncoupled and uncorrelated, so all these factors are in a probability sense completely independent, then

$Z = Z_t Z_c Z_n Z_e Z_r Z_v,$

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.

Nuclear	$Z_n = 1 qquad (T < 10^8 K)$
Electronic	$Z_e = W_0 exp(kT D_e + W_1 exp(-theta_{e1}/T) + cdots)$
Vibrational	$Z_v = prod_j frac{exp(-theta_{vj} / 2T)}{1 - exp(-theta_{vj} / T)}$
Rotational (linear)	$Z_r = frac{T}{sigma} theta_r$
Rotational (non-linear)	$Z_r = frac{1}{sigma}sqrt{frac{{pi}T^3}{theta_A theta_B theta_C}}$
Translational	$Z_t = frac{(2 pi mkT)^{3/2}}{h^3}$
Configurational (ideal gas)	$Z_c = V,$

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:

$P = P_t + P_c + P_n + P_e + P_r + P_v,$

Grand canonical ensemble

Main article: Grand canonical ensemble

If the system under study is an open system, (matter can be exchanged), and particle number is conserved, we would have to introduce chemical potentials, μ_j, j=1,...,n and replace the canonical partition function with the grand canonical partition function: In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ... In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a... In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ... In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ...

$Xi(V,T,mu) = sum_i expleft(beta left[sum_{j=1}^n mu_j N_{ij}-E_iright ]right)$

where N_ij is the number of j^th species particles in the i^th configuration. Sometimes, we also have other variables to add to the partition function, one corresponding to each conserved quantity. Most of them, however, can be safely interpreted as chemical potentials. In most condensed matter systems, things are nonrelativistic and mass is conserved. However, most condensed matter systems of interest also conserve particle number approximately (metastably) and the mass (nonrelativistically) is none other than the sum of the number of each type of particle times its mass. Mass is inversely related to density, which is the conjugate variable to pressure. For the rest of this article, we will ignore this complication and pretend chemical potentials don't matter. See grand canonical ensemble. In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ...

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, j is the index for those particles of species j and i is the index for microstate i:

$U = sum_i E_i frac{exp(-beta (E_i-sum_j mu_j N_{ij}))}{Xi}$

$N_j = sum_i N_{ij} frac{exp(-beta (E_i-sum_j mu_j N_{ij}))}{Xi}$

Grand potential:	$Phi_{G} = - {ln Xiover beta}$
Internal energy:	$U = -left( frac{partialln Xi}{partialbeta} right)_{mu}+sum_i{mu_ioverbeta}left({partial ln Xiover partial mu_i}right )_{beta}$
Particle number:	$N_i={1overbeta}left({partial ln Xiover partial mu_i}right)_beta$
Entropy:	$S = k (ln Xi + beta U- beta sum_i mu_i N_i),$
Helmholtz free energy:	$F = G+sum_i mu_i N_i=-{ln Xiover beta} +sum_i{mu_iover beta} left( frac{partial ln Xi}{partial mu_i}right)_{beta}$

There are very few or no other articles that link to this one. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the â€œusefulâ€ work obtainable from a closed thermodynamic system at a constant temperature. ...

Equivalence between descriptions at the thermodynamic limit

All the above descriptions differ in the way they allow the given system to fluctuate between its configurations. In physics and physical chemistry, the thermodynamic limit is reached as the number of particles (atoms or molecules) in a system N approaches infinity â€” or in practical terms, one mole or Avogadros number â‰ˆ 6 x 1023. ...

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 physics, heat, symbolized by Q, is defined as transfer of thermal energy [1] Generally, heat is a form of energy transfer associated with the different motions of atoms, molecules and other particles that comprise matter when it is hot and when it is cold. ...

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. In physics and physical chemistry, the thermodynamic limit is reached as the number of particles (atoms or molecules) in a system N approaches infinity â€” or in practical terms, one mole or Avogadros number â‰ˆ 6 x 1023. ...

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 walkers

The 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, $t' = i t$ . A polymer is a long, repeating chain of atoms, formed through the linkage of many molecules called monomers. ...

Random walks in time

The 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 b, depending on whether a coin lands heads or tails when flipped. Lets start by considering the statistics of the steps the toy train takes (where $S i$ is the ith step taken):

$langle S_{i} rangle = 0$ ; due to a priori equal probabilities
$langle S_{i} S_{j} rangle = b^2 delta_{ij}$

The second quantity is known as the correlation function. The delta is the kronecker delta which tells us that if the indices i and j are different, then the result is 0, but if i = j then the kronecker delta is 1, so the correlation function returns a value of $b 2$ . This makes sense, because if i = j then we are considering the same step. Rather trivially then it can be shown that the average displacement of the train from the x-axis is 0; For stochastic processes, including those that arise in statistical mechanics and Euclidean quantum field theory, a correlation function is the correlation between random variables at two different points in space or time. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... For stochastic processes, including those that arise in statistical mechanics and Euclidean quantum field theory, a correlation function is the correlation between random variables at two different points in space or time. ...

$x = sum_{i=1}^{N} S_{i}$
$langle x rangle = langle sum_{i=1}^{N} S_{i} rangle$
$langle x rangle = sum_{i=1}^{N} langle S_{i} rangle$

As stated $langle S_{i} rangle$ 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

$x_{rms} = sqrt {langle x^2 rangle} = b sqrt N$

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 N is simply the number of steps moved (is loosely connected with time) and b is the characteristic step length. As a consequence we can consider diffusion as a random walk process. The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ...

Random walks in space

Random 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: self-avoiding random walks, where the links of the polymer chain interact and do not overlap in space, and pure random walks, where the links of the polymer chain are non-interacting and links are free to lie on top of one another. The former type is most applicable to physical systems, but their solutions are harder to get at from first principles.

By considering a freely jointed, non-interacting polymer chain, the end-to-end vector is $mathbf{R} = sum_{i=1}^{N} mathbf r_i$ where $mathbf {r}_{i}$ is the vector position of the i-th link in the chain. As a result of the central limit theorem, if N >> 1 then the we expect a Gaussian distribution for the end-to-end vector. We can also make statements of the statistics of the links themselves;
$langle mathbf{r}_{i} rangle = 0$ ; by the isotropy of space
$langle mathbf{r}_{i} cdot mathbf{r}_{j} rangle = b^2 delta_{ij}$ ; all the links in the chain are uncorrelated with one another
Using the statistics of the individual links, it is easily shown that $langle mathbf R rangle = 0$ and $langle mathbf R cdot mathbf R rangle = Nb^2$ . Notice this last result is the same as that found for random walks in time. A central limit theorem is any of a set of weak-convergence results in probability theory. ... Probability density function of Gaussian distribution (bell curve). ...

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

$P = frac{1}{left (frac{2 pi N b^2}{3} right )^{3/2}} exp frac {-3 mathbf R cdot mathbf R}{2NB^2}$

What use is this to us? Recall that according to the principle of equally likely a priori probabilities, the number of microstates, Ω, at some physical value is directly proportional to the probability distribution at that physical value, viz;

$Omega left ( mathbf{R} right ) = c Pleft ( mathbf{R} right )$

where c is an arbitrary proportionality constant. Given our distribution function, there is a maxima corresponding to $mathbf {R} = 0$ . Physically this amounts to there being more microstates which have an end-to-end vector of 0 than any other microstate. Now by considering

$S left ( mathbf {R} right ) = k_B ln Omega {left ( mathbf R right) }$
$Delta S left( mathbf {R} right ) = S left( mathbf {R} right ) - S left (0 right )$
$Delta F = - T Delta S left ( mathbf {R} right )$

where F is the Helmholtz free energy it is trivial to show that

$Delta F = k_B T frac {3R^2}{2Nb^2} = frac {1}{2} K R^2 quad ; K = frac {3 k_B T}{Nb^2}$

A Hookian spring!
This result is known as the Entropic Spring Result and amounts to saying that upon stretching a polymer chain you are doing work on the system to drag it away from its (preferred) equilibrium state. An example of this is a common elastic band, composed of long chain (rubber) polymers. By stretching the elastic band you are doing work on the system and the band behaves like a conventional spring. What is particularly astonishing about this result however, is that the work done in stretching the polymer chain can be related entirely to the change in entropy of the system as a result of the stretching. In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the â€œusefulâ€ work obtainable from a closed thermodynamic system at a constant temperature. ...

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

A list of notable textbooks in statistical mechanics, arranged by date. ...

External links

Philosophy of Statistical Mechanics article by Lawrence Sklar for the Stanford Encyclopedia of Philosophy.
SklogWiki - a statistical mechanics wiki.

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General subfields within physics

Classical mechanics · Electromagnetism · Thermodynamics · General relativity · Quantum mechanics The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ... Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the fundamental laws of the universe. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Electromagnetism is the physics of the electromagnetic field; a field encompassing all of space which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Fig. ...

Particle physics · Thermal physics · Condensed matter physics · Atomic, molecular, and optical physics Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Thermal physics is the combined study of thermodynamics, statistical mechanics, and kinetic theory. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Atomic, molecular, and optical physics is the study of matter-matter and light-matter interactions on the scale of single atoms or structures containing a few atoms. ...

Categories: Fundamental physics concepts | Statistical mechanics | Mechanics | Thermodynamics

Results from FactBites:

Statistical mechanics - Wikipedia, the free encyclopedia (2460 words)
	Statistical mechanics is the application of statistics, 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.
	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.
	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.

statistical mechanics. The Columbia Encyclopedia, Sixth Edition. 2001-05 (414 words)
	The foundations of statistical mechanics can be traced to the 19th-century work of Ludwig Boltzmann, and the theory was further developed in the early 20th cent.
	Maxwell-Boltzmann statistics apply to systems of classical particles, such as the atmosphere, in which considerations from the quantum theory are small enough that they may be ignored.
	Statistical mechanics has also yielded deep insights in the understanding of magnetism, phase transitions, and superconductivity.

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