ميكانيكا إحصائية

الميكانيكا الإحصائية هو تطبيق علم الاحصاء, الذي يتألف من مجموعة أدوات رياضية للتعامل مع التجمعات الضخمة, ضمن مجال الميكانيك الذي يهتم بحركة الجسيمات أو الأجسام عند خضوعها لقوى خارجية. لذلك يؤمن الميكانيك الإحصائي إطارا لربط الخواص المجهرية properties microscopic للذرات والجزيئات مع الخواص الظاهرة (الجهرية) macroscopic properties للمواد المدروسة. فهي تقوم بتفسير التحريك الحراري على أنه نتيجة للإحصاء مع الميكانيك بجانبيه (الكلاسيكي والكمومي).

الميكانيكا الإحصائية هو تطبيق نظريات الإحصاء التي تتضمن أدوات رياضية للتعامل مع التجمعات الكبيرة ، في فروع الفيزياء التي تتعامل مع حركة أعداد كبيرة من الأجسام أو الجزيئات عند تعريضها لقوى معينة .

يشكل الميكانيك الإحصائي اطارا يربط الخواص المجهرية Microscopic Properties للجزيئات مع الخواص الجهرية Macroscopic Properties للمواد التي تتألف أساسا من هذه الجزيئات مما يعطينا فكرة جيدة عن أصل الخواص المواد التي نراها يوميا في الحياة العادية .

أحد أهم فروعه هو التحريك الحراري ( الترموديناميك Thermodynamics ) الذي يعتبر نتيجة لعلمي الإحصاء و الميكانيك ( الكلاسيكي منه و الكمومي ) .

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 الصلة بالترموديناميكا

${\displaystyle{\displaystyle\langle E\rangle={\sum_{i}E_{i}e^{-\beta E_{i}}\over Z}=-{1\over Z}{dZ\over d\beta}}}$

implies, together with the interpretation of ${\displaystyle{\displaystyle\langle E\rangle}}$ as ${\displaystyle{\displaystyle U\ }}$, the following microscopic definition of internal energy:

${\displaystyle{\displaystyle U\colon=-{d\ln Z\over d\beta}.}}$

The entropy can be calculated by (see Shannon entropy)

${\displaystyle{\displaystyle{S\over k}=-\sum_{i}p_{i}\ln p_{i}=\sum_{i}{e^{-\beta E_{i}}\over Z}(\beta E_{i}+\ln Z)=\ln Z+\beta U}}$

which implies that

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

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

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

${\displaystyle{\displaystyle\langle J\rangle=\sum_{i}p_{i}J_{i}=\sum_{i}J_{i}{\frac{e^{-\beta E_{i}}}{Z}}}}$

where ${\displaystyle{\displaystyle\langle J\rangle}}$ is the average value of property ${\displaystyle{\displaystyle J\ }}$. This equation can be applied to the internal energy, ${\displaystyle{\displaystyle U\ }}$:

${\displaystyle{\displaystyle U=\sum_{i}E_{i}{\frac{e^{-\beta E_{i}}}{Z}}}}$

Helmholtz free energy:	${\displaystyle{\displaystyle F=-{\ln Z\over\beta}}}$
Internal energy:	${\displaystyle{\displaystyle U=-\left({\frac{\partial\ln Z}{\partial\beta}}\right)_{N,V}}}$
Pressure:	${\displaystyle{\displaystyle P=-\left({\partial F\over\partial V}\right)_{N,T}={1\over\beta}\left({\frac{\partial\ln Z}{\partial V}}\right)_{N,T}}}$
Entropy:	${\displaystyle{\displaystyle S=k(\ln Z+\beta U)\,}}$
Gibbs free energy:	${\displaystyle{\displaystyle G=F+PV=-{\ln Z\over\beta}+{V\over\beta}\left({\frac{\partial\ln Z}{\partial V}}\right)_{N,T}}}$
Enthalpy:	${\displaystyle{\displaystyle H=U+PV\,}}$
Constant Volume Heat capacity:	${\displaystyle{\displaystyle C_{V}=\left({\frac{\partial U}{\partial T}}\right)_{N,V}}}$
Constant Pressure Heat capacity:	${\displaystyle{\displaystyle C_{P}=\left({\frac{\partial H}{\partial T}}\right)_{N,P}}}$
Chemical potential:	${\displaystyle{\displaystyle\mu_{i}=-{1\over\beta}\left({\frac{\partial\ln Z}{\partial N_{i}}}\right)_{T,V,N}}}$

${\displaystyle{\displaystyle E=E_{t}+E_{c}+E_{n}+E_{e}+E_{r}+E_{v}\,}}$

Nuclear	${\displaystyle{\displaystyle Z_{n}=1\qquad(T<10^{8}K)}}$
Electronic	${\displaystyle{\displaystyle Z_{e}=W_{0}e^{kTD_{e}+W_{1}e^{-\theta_{e1}/T}+\cdots}}}$
Vibrational	${\displaystyle{\displaystyle Z_{v}=\prod_{j}{\frac{e^{-\theta_{vj}/2T}}{1-e^{-\theta_{vj}/T}}}}}$
Rotational (linear)	${\displaystyle{\displaystyle Z_{r}={\frac{T}{\sigma}}\theta_{r}}}$
Rotational (non-linear)	${\displaystyle{\displaystyle Z_{r}={\frac{1}{\sigma}}{\sqrt{\frac{{\pi}T^{3}}{\theta_{A}\theta_{B}\theta_{C}}}}}}$
Translational	${\displaystyle{\displaystyle Z_{t}={\frac{(2\pi mkT)^{3/2}}{h^{3}}}}}$
Configurational (ideal gas)	${\displaystyle{\displaystyle Z_{c}=V\,}}$

 Grand canonical ensemble

المقالة الرئيسية: Grand canonical ensemble

If the system under study is an open system, (matter can be exchanged), but particle number is not 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:

${\displaystyle{\displaystyle\Xi(V,T,\mu)=\sum_{i}\exp\left(\beta\left[\sum_{j=1}^{n}\mu_{j}N_{ij}-E_{i}\right]\right)}}$

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.

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:

${\displaystyle{\displaystyle U=\sum_{i}E_{i}{\frac{\exp(-\beta(E_{i}-\sum_{j}\mu_{j}N_{ij}))}{\Xi}}}}$

${\displaystyle{\displaystyle N_{j}=\sum_{i}N_{ij}{\frac{\exp(-\beta(E_{i}-\sum_{j}\mu_{j}N_{ij}))}{\Xi}}}}$

Grand potential:	${\displaystyle{\displaystyle\Phi_{G}=-{\ln\Xi\over\beta}}}$
Internal energy:	${\displaystyle{\displaystyle U=-\left({\frac{\partial\ln\Xi}{\partial\beta}}\right)_{\mu}+\sum_{i}{\mu_{i}\over\beta}\left({\partial\ln\Xi\over\partial\mu_{i}}\right)_{\beta}}}$
Particle number:	${\displaystyle{\displaystyle N_{i}={1\over\beta}\left({\partial\ln\Xi\over\partial\mu_{i}}\right)_{\beta}}}$
Entropy:	${\displaystyle{\displaystyle S=k(\ln\Xi+\beta U-\beta\sum_{i}\mu_{i}N_{i})\,}}$
Helmholtz free energy:	${\displaystyle{\displaystyle F=G+\sum_{i}\mu_{i}N_{i}=-{\ln\Xi\over\beta}+\sum_{i}{\mu_{i}\over\beta}\left({\frac{\partial\ln\Xi}{\partial\mu_{i}}}\right)_{\beta}}}$

 انظر أيضاً

A Table of Statistical Mechanics Articles

	Maxwell Boltzmann	Bose-Einstein	Fermi-Dirac
Particle		Boson	Fermion
Statistics	Partition function Identical particles#Statistical properties Microcanonical ensemble | Canonical ensemble | Grand canonical ensemble
Statistics	Maxwell-Boltzmann statistics Maxwell-Boltzmann distribution Boltzmann distribution Gibbs paradox	Bose-Einstein statistics	Fermi-Dirac statistics
Thomas-Fermi approximation	gas in a box gas in a harmonic trap
Gas	Ideal gas	Bose gas Debye model Bose-Einstein condensate Planck's law of black body radiation	Fermi gas Fermion condensate
Chemical Equilibrium	Classical Chemical equilibrium

 مصادر

• Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics. 2nd ed. John Wiley & Sons, Inc., New York. ISBN 0-471-86256-8
• P. Pluch Quantum Probability Theory, PhD Thesis, University of Klagenfurt (2006)

 وصلات خارجية

• Philosophy of Statistical Mechanics article by Lawrence Sklar for the Stanford Encyclopedia of Philosophy.