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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Statistical mechanics and kinetic theory}}
'''Quantum statistical mechanics''' is the branch of physics that explains how macroscopic thermodynamic behavior emerges from the collective properties of many quantum particles.<ref name="Huang">{{cite book
|last=Huang
|first=Kerson
|title=Statistical Mechanics
|edition=2
|publisher=John Wiley & Sons
|year=1987
|isbn=0471815187
}}</ref><ref name="LandauLifshitz">{{cite book
|last1=Landau
|first1=L. D.
|last2=Lifshitz
|first2=E. M.
|title=Statistical Physics, Part 1
|edition=3
|publisher=Butterworth-Heinemann
|year=1980
|isbn=9780750633727
}}</ref> It provides the conceptual link between microscopic [[Physics:Quantum mechanics|quantum mechanics]] and macroscopic thermodynamics, and is fundamental for the theory of many-body systems, [[Physics:Quantum Partition function|partition functions]], [[Physics:Quantum Distribution functions|quantum distribution functions]], superconductivity, and superfluidity.<ref name="KadanoffBaym">{{cite book
|last1=Kadanoff
|first1=Leo P.
|last2=Baym
|first2=Gordon
|title=Quantum Statistical Mechanics
|publisher=CRC Press
|year=2018
|isbn=9780201410464
}}</ref><ref name="Pathria">{{cite book
|last1=Pathria
|first1=R. K.
|last2=Beale
|first2=Paul D.
|title=Statistical Mechanics
|edition=3
|publisher=Elsevier
|year=2011
|isbn=9780123821881
}}</ref>
[[File:Quantaum statistical mechanica visualisation-1.jpg|thumb|300px|Conceptual illustration of quantum statistical mechanics, bridging microscopic quantum states and macroscopic statistical behavior]]

==From quantum states to ensembles==
In ordinary quantum mechanics, a system is described by a state vector <math>|\psi\rangle</math>. For macroscopic systems with enormous numbers of degrees of freedom, however, it is usually neither practical nor physically appropriate to specify a single pure state in complete detail.<ref name="Huang"/><ref name="Peres">{{cite book
|last=Peres
|first=Asher
|title=Quantum Theory: Concepts and Methods
|publisher=Kluwer
|year=1993
|isbn=0792325494
}}</ref>

Instead, one introduces '''statistical ensembles''' that describe probabilities for different possible quantum states consistent with the macroscopic constraints on the system, such as fixed temperature, energy, or particle number.<ref name="LandauLifshitz"/><ref name="Reichl">{{cite book
|last=Reichl
|first=Linda E.
|title=A Modern Course in Statistical Physics
|edition=4
|publisher=Wiley
|year=2016
|isbn=9783527413492
}}</ref>

==Density matrix formalism==
The natural language of quantum statistical mechanics is the '''density operator''' or '''density matrix'''. A general state is written as

<math>
\rho = \sum_i p_i |\psi_i\rangle \langle \psi_i|,
</math>

where the coefficients <math>p_i</math> are probabilities satisfying <math>p_i \ge 0</math> and <math>\sum_i p_i = 1</math>.<ref name="Fano1957">{{cite journal
|last=Fano
|first=U.
|title=Description of States in Quantum Mechanics by Density Matrix and Operator Techniques
|journal=Reviews of Modern Physics
|year=1957
|volume=29
|issue=1
|pages=74–93
|doi=10.1103/RevModPhys.29.74
|url=https://link.aps.org/doi/10.1103/RevModPhys.29.74
}}</ref>

Expectation values of observables are given by

<math>
\langle A \rangle = \mathrm{Tr}(\rho A).
</math>

This formalism includes both pure states and mixed states, and is indispensable for describing thermal equilibrium, open systems, and subsystems of larger entangled systems.<ref name="Peres"/><ref name="NielsenChuang">{{cite book
|last1=Nielsen
|first1=Michael A.
|last2=Chuang
|first2=Isaac L.
|title=Quantum Computation and Quantum Information
|edition=10th anniversary
|publisher=Cambridge University Press
|year=2010
|isbn=9780521635035
}}</ref>

==Statistical ensembles==
Different physical constraints lead to different equilibrium ensembles.<ref name="LandauLifshitz"/><ref name="Huang"/>

===Canonical ensemble===
For a system in contact with a heat bath at temperature <math>T</math>, the density operator is

<math>
\rho = \frac{e^{-\beta H}}{Z},
</math>

where

<math>
\beta = \frac{1}{k_B T}
</math>

and

<math>
Z = \mathrm{Tr}(e^{-\beta H})
</math>

is the canonical partition function.<ref name="Huang"/><ref name="KadanoffBaym"/>

If the Hamiltonian has eigenvalues <math>E_n</math>, then

<math>
Z = \sum_n e^{-\beta E_n}.
</math>

The probability of occupying a state with energy <math>E_n</math> is proportional to its Boltzmann weight,

<math>
P_n = \frac{e^{-\beta E_n}}{Z}.
</math>

===Grand canonical ensemble===
For open systems that can exchange both energy and particles with a reservoir, the appropriate ensemble is the grand canonical ensemble:

<math>
\rho = \frac{e^{-\beta(H-\mu N)}}{\mathcal{Z}},
</math>

with grand partition function

<math>
\mathcal{Z} = \mathrm{Tr}\left(e^{-\beta(H-\mu N)}\right).
</math>

Here <math>\mu</math> is the chemical potential and <math>N</math> is the particle-number operator.<ref name="Pathria"/><ref name="Reichl"/>

This formulation is essential for quantum gases, photons, phonons, electrons in solids, and many-body field-theoretic treatments.<ref name="KadanoffBaym"/><ref name="LandauLifshitz"/>

==Partition function==
The partition function is the generating quantity from which the thermodynamic properties of the system can be derived.<ref name="Pathria"/><ref name="Huang"/> For the canonical ensemble,

<math>
Z = \mathrm{Tr}(e^{-\beta H}).
</math>

The internal energy is

<math>
U = -\frac{\partial \ln Z}{\partial \beta},
</math>

the Helmholtz free energy is

<math>
F = -k_B T \ln Z,
</math>

and the entropy can be obtained from

<math>
S = -\left(\frac{\partial F}{\partial T}\right)_V.
</math>

Thus the partition function encodes the full equilibrium thermodynamics of the system.<ref name="LandauLifshitz"/><ref name="Pathria"/>

==Quantum statistics of identical particles==
A central feature of quantum statistical mechanics is that identical particles are fundamentally indistinguishable.<ref name="Huang"/> This leads to two basic types of quantum statistics:

===Bosons===
Bosons obey Bose-Einstein statistics, with mean occupation number

<math>
n(\epsilon) = \frac{1}{e^{\beta(\epsilon-\mu)} - 1}.
</math>

This allows multiple particles to occupy the same state, leading to phenomena such as Bose-Einstein condensation and superfluidity.<ref name="Pathria"/><ref name="Kardar">{{cite book
|last=Kardar
|first=Mehran
|title=Statistical Physics of Particles
|publisher=Cambridge University Press
|year=2007
|isbn=9780521873420
}}</ref>

===Fermions===
Fermions obey Fermi-Dirac statistics, with mean occupation number

<math>
n(\epsilon) = \frac{1}{e^{\beta(\epsilon-\mu)} + 1}.
</math>

Because of the Pauli exclusion principle, no more than one fermion can occupy a single-particle quantum state, which is crucial for the behavior of electrons in atoms, metals, and degenerate matter.<ref name="Pathria"/><ref name="LandauLifshitz"/>

==Entropy==
The entropy of a quantum system is given by the '''von Neumann entropy''':

<math>
S = -k_B \mathrm{Tr}(\rho \ln \rho).
</math>

If the density operator is diagonalized as

<math>
\rho = \sum_j \eta_j |j\rangle \langle j|,
</math>

then the entropy becomes

<math>
S = -k_B \sum_j \eta_j \ln \eta_j.
</math>

This is the direct quantum generalization of Gibbs entropy and plays a central role in thermal physics, information theory, and the study of entanglement.<ref name="Peres"/><ref name="NielsenChuang"/>

==Thermal equilibrium and maximum entropy==
Equilibrium ensembles in quantum statistical mechanics can be understood through the principle of maximum entropy. Among all density operators consistent with specified macroscopic constraints, the physical equilibrium state is the one that maximizes the von Neumann entropy.<ref name="Peres"/><ref name="Reichl"/>

For fixed average energy this gives the canonical ensemble, while fixing both average energy and average particle number yields the grand canonical ensemble.<ref name="Reichl"/>

==Emergence of classical behavior==
In many-particle systems, interactions, coarse graining, and coupling to an environment suppress observable quantum coherence and make classical statistical descriptions effective.<ref name="Zurek">{{cite journal
|last=Zurek
|first=W. H.
|title=Decoherence, einselection, and the quantum origins of the classical
|journal=Reviews of Modern Physics
|year=2003
|volume=75
|issue=3
|pages=715–775
|doi=10.1103/RevModPhys.75.715
|url=https://link.aps.org/doi/10.1103/RevModPhys.75.715
}}</ref>

This does not mean the underlying theory stops being quantum; rather, classical thermodynamic behavior emerges as an effective description of large systems with inaccessible microscopic details.<ref name="LandauLifshitz"/><ref name="Zurek"/>

==Connection to kinetic theory==
Quantum statistical mechanics provides the equilibrium and near-equilibrium foundation for [[Physics:Quantum Kinetic theory|quantum kinetic theory]]. At larger scales and away from equilibrium, one often uses distribution functions such as

<math>
f(\mathbf{x},\mathbf{p},t),
</math>

which satisfy transport equations like the Boltzmann equation or more general quantum kinetic equations.<ref name="Cercignani">{{cite book
|last=Cercignani
|first=C.
|title=The Boltzmann Equation and Its Applications
|publisher=Springer
|year=1988
|isbn=9780387963464
}}</ref><ref name="Bonitz">{{cite book
|last=Bonitz
|first=Michael
|title=Quantum Kinetic Theory
|publisher=Teubner
|year=1998
|isbn=9783519002540
}}</ref>

In this sense, quantum statistical mechanics describes equilibrium ensembles and thermodynamic structure, while kinetic theory extends the description to time-dependent nonequilibrium evolution.<ref name="KadanoffBaym"/><ref name="Bonitz"/>

==Applications==
Quantum statistical mechanics is essential in:

* ideal Bose and Fermi gases
* black-body radiation
* condensed-matter systems
* superconductivity and superfluidity
* quantum information and entanglement theory
* many-body localization and thermalization studies<ref name="Pathria"/><ref name="NandkishoreHuse2015">{{cite journal
|last1=Nandkishore
|first1=Rahul
|last2=Huse
|first2=David A.
|title=Many-Body Localization and Thermalization in Quantum Statistical Mechanics
|journal=Annual Review of Condensed Matter Physics
|year=2015
|volume=6
|pages=15–38
|doi=10.1146/annurev-conmatphys-031214-014726
|url=https://arxiv.org/abs/1404.0686
}}</ref>

==Physical interpretation==
'''Quantum statistical mechanics explains how'''

* microscopic quantum states are converted into ensemble descriptions
* partition functions generate thermodynamic quantities
* indistinguishable particles produce Bose-Einstein and Fermi-Dirac statistics
* entropy and equilibrium emerge in many-particle quantum systems
* macroscopic thermodynamics arises from underlying quantum laws<ref name="Huang"/><ref name="LandauLifshitz"/><ref name="Pathria"/>

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}
{{Author|Harold Foppele}}
[[Category:Quantum mechanics]]
[[Category:Statistical mechanics]]
[[Category:Quantum mechanical entropy]]

{{Sourceattribution|Quantum Statistical mechanics|1}}

Physics:Quantum Statistical mechanics - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template