The partition function describes the statistical properties of a system in thermodynamic equilibrium, and is constructed to represent a particular statistical ensemble (microcanonical, macrocanonical, grand canonical).
Questions tagged [partition-function]
658 questions
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How exact is the analogy between statistical mechanics and quantum field theory?
Famously, the path integral of quantum field theory is related to the partition function of statistical mechanics via a Wick rotation and there is therefore a formal analogy between the two. I have a few questions about the relation between the two…
user26866
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Can Lee-Yang zeros theorem account for triple point phase transition?
Now the prominent Lee-Yang theorem (or Physical Review 87, 410, 1952) has almost become a standard ingredient of any comprehensive statistical mechanics textbook.
If the volume tends to infinity, some complex roots of the grand canonical partition…
xiaohuamao
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55
votes
4 answers
The unreasonable effectiveness of the partition function
In a first course on statistical mechanics the partition function is normally introduced as the normalisation for the probability of a particle being in a particular energy level.
$$p_j=\frac{1}{Z}\exp\left(\frac{-E_j}{k_bT}\right),$$
$$Z=\sum_j…
ChrisM
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Why is the partition function called ''partition function''?
The partition function plays a central role in statistical mechanics.
But why is it called ''partition function''?
Arnold Neumaier
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votes
7 answers
Why is the contribution of a path in Feynmans path integral formalism $\sim e^{(i/\hbar)S[x(t)]}$?
In the book "Quantum Mechanics and Path Integrals" Feynman & Hibbs state that
The probability $P(b,a)$ to go from point $x_a$ at the time $t_a$ to the point $x_b$ at the time $t_b$ is the absolute square $P(b,a) = \|K(b,a)\|^2$ of an amplitude…
asmaier
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3 answers
What is the physical meaning of the partition function in statistical physics?
In many places in statistical physics we use the partition function. To me, the explanations of their use are clear, but I wonder what their physical significance is. Can anyone please explain with a good example without too many mathematical…
albedo
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votes
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Why is the partition function divided by $(h^{3N} N!)$?
When computing partition functions for classical systems with $N$ particles with a given Hamiltonian $H$ I've seen some places writing it as
$$Z = \dfrac{1}{h^{3N} N!}\int e^{-\beta H(p,q)}dpdq$$
where the integral is over the available phase space.…
Gold
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Why is the canonical partition function the Laplace transform of the microcanonical partition function?
This web page says that the microcanonical partition function
$$
\Omega(E) = \int \delta(H(x)-E) \,\mathrm{d}x
$$
and the canonical partition function
$$
Z(\beta) = \int e^{-\beta H(x)}\,\mathrm{d}x
$$
are related by the fact that $Z$ is the Laplace…
Casey Chu
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3 answers
Is there a way to obtain the classical partition function from the quantum partition function in the limit $h \rightarrow 0$?
One would like to motivate the classical partition function in the following way: in the limit that the spacing between the energies (generally on the order of $h$) becomes small relative to the energies themselves, one might…
Pricklebush Tickletush
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21
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1 answer
Continuum Field Theory for the Ising Model
My problem is to take the $d$-dimensional Ising Hamiltonian,
$$H = -\sum_{i,j}\sigma_i J_{i,j} \sigma_j - \sum_{i} \tilde{h}_i \sigma_i$$
where $J_{ij}$ is a matrix describing the couplings between sites $i$ and $j$. Applying a Hubbard-Stratonovich…
Kai
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How to turn a sum into an integral?
In, An Introduction to Thermal Physics, page 235, Schroder wants to evaluate the partition function
$$Z_{tot}=\sum_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}$$
in the limit that $kT\gg\epsilon$, thus he writes
$$Z_{tot}\approx\int_0^\infty…
GedankenExperimentalist
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votes
3 answers
What is the Spectral Form Factor?
In many papers in Random Matrix Theory [1-3] related to quantum chaos (and, in particular, to the SYK model) they analytically continuate the partition function of the system $Z(\beta)$ into $Z(\beta + it)$ and then define the Spectral Form Factor…
P. C. Spaniel
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Partition function of a hydrogen gas
Hi I have a doubt (I'm not very expert in statistical mechanics, so sorry for this question). We consider a gas of hydrogen atoms with no interactions between them.
The partition function is:
$$
Z=\frac{Z_s^N}{N!}
$$
where $Z_s$ is the partition…
Boy S
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Measure of Lee-Yang zeros
Consider a statistical mechanical system (say the 1D Ising model) on a finite lattice of size $N$, and call the corresponding partition function (as a function of, say, real temperature and real magnetic field) $Z^{(N)}(t, h)$, where $t$ is…
user3657
15
votes
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About variational methods, renormalization and $a$, $c$-theorems
Variational approximation
Variational methods are an important technique, frequently applied for the approximation of complicated probability distributions, with the applications in statistical physics and data science.
Suppose we have some…
spiridon_the_sun_rotator
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