Questions tagged [partition-function]
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).
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Coherent states and thermal properties
I am reading a paper called Thermodynamics of Coherent States and Black Hole Entropy, written by Bashkirov and Sukhanov. If I understand correctly, they define a coherent state by the equation
$$a|d\...
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Exercise 12.4 Charles Kittel Elementary Statistical Physics
The text says:
Consider a monoatomic crystal consisting of $N$ atoms. These may be situate in two kinds of position: normal and interstitial. The energy of an atom in the interstitial position is ...
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Partition function for dipolar dielectrics
How can you write the partition function for non-polar and polar dielectric materials? For dipolar dielectric materials we can easily write partition function; it is proportional to $\exp(-\beta\mu ...
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Summing graphs in the partition function (statistical physics)
I am looking at Tong's lecture notes on statistical physics, and I wanted to understand a step in his cluster expansion better.
The goal here is to calculate the partition function in the canonical ...
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Independence of Trace-Partition function
I am trying to calculate the partition function of the system of two completely decoupled systems. Probability-wise, the decoupled nature means that the PDF is the product of the PDF of each subsystem....
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Free energy of a one-dimensional harmonic oscillator
The potential energy of a one-dimensional harmonic potential can be expressed as $U(x)=\frac{1}{2}K(x-x_0)^2$, where $K$ is the force constant and $x_0$ is the equilibrium position. I'm wondering how ...
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Are Distribution functions really the probability or the number of particles?
I am studying the distribution functions in statistical mechanics (Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac).
Are these distribution functions give the number of particles in an energy level ...
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Dimensional inconsistency in evaluating the canonical partition function
We know that canonical partition of an $N$-particle system is given as
$$Z=\!\!\!\!\!\!\!\!\!\!\!\!\sum_{\text{All possible microstates}}\!\!\!\!\!\!\!\!\!\!\!\!e^{-\beta E}=\sum_E\Omega(E)e^{-\beta E}...
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Changing variable in path integral
Good evening,
I am learning about path integrals in QFT and I was wondering, can you simplify the path integral by shifting the fields? To make it more clear I will give you an example. Suppose that I ...
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Why doesn't this diagram appear in the partition function in zero-dimensional QFT?
For the zero-dimensional QFT with action
$$S(\phi)=\frac{\alpha}{2}\phi^2+\frac{\lambda}{4!}\phi^4-J\phi,\tag{1}$$
we can perturbatively expand the partition function as
$$Z_\lambda(J)=\int_{-\infty}^{...
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Free boson twisted boundary condition and $T^2$ partition function
Many CFT textbooks discuss free boson theory and free fermion theories on the torus.
The partition function for the boson theory (without compactification and orbifold) is obtained by summing over the ...
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Calculating $\langle\hat{\phi_i}\rangle_t$ (Blundell's Quantum field theory) (EDITED) [closed]
I am reading Blundell's Quantum field theory for the Gifted Amateur and stuck at some calculation. In his book p.197, 21.2 Sources in statistical physics, he defined the partition function with the ...
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On Quantum Harmonic Oscillator in Canonical Ensemble [duplicate]
I'm a bit confused with some results from the interpretation of the QHO in a canonical ensemble.
The partition function is given by the expression
$$Z = \sum_{s} e^{-\beta E_s},$$
where $s$ represents ...
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How to express internal energy in terms of the canonical function summing over energy levels?
as far as I understand, the canonical partition function of a single particle can be expressed as follows:
$$ z = \sum_i e^{-\beta\cdot\epsilon_i} $$
Where $i$ are the micro states, $\beta$ is the ...
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Partition function for a 1 dimensional polymer
A one-dimensional polymer (a chain), made of (N + 1) monomers, is diffusing on top of a one-dimensional lattice having a lattice constant a = 1. The i-th monomer (i = 0, 1, 2,..., N) is located at a ...
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Discrepancy between energy calculated via the equipartition theorem and the partition function
I am doing some homework on statistical mechanics, and I'm facing something I can't seem to understand/find what I'm doing wrong.
Suppose we have a diatomic molecule, both atoms have mass $m$. Suppose ...
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Generating functional for scalar expectation values in String Theory (Polchinski 6.2, Volume 1)
Polchinski defines generating functional as:
$$\begin{align*}
Z[J] = \left\langle\exp{\left[i \int d^2\, \sigma J(\sigma) \cdot X(\sigma)\right]}\right\rangle
\end{align*}\tag{6.2.1}$$
for the ...
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How to connect the usage of the path integral in QFT to the usage in Quantum Mechanics?
In Quantum Mechanics, path integrals are used to calculate the matrix element:
$$
\langle x_1, t_1|x_2, t_2\rangle_J=\int
e^{i(S[x(t)]+\int\!J x(t))/\hbar} d[x(t) ].\tag{1}$$
If we naively try to ...
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Factorization of 1d Ising model partition function
If I'm studying a 1-dimensional Ising model such that $\mathcal H = \sum_k J_k\sigma_k\sigma_{k+1}$, where $$J_k=\begin{cases}J&k \in2\mathbb N\\2J&k\in2\mathbb N+1 \end{cases}$$ can I ...
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Quantum Mapping of a Generalized 2D XY Model
I have been reading this paper, which discusses the deconfinement transition in the 2D generalized XY model. The classical Hamiltonian is given by the following. See page 2 of the paper.
$$
H = - J\...
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Does a CFT need a UV regulator?
I have a very basic question about conformal field theory: Does the partition function need a UV regulator, or is it finite even without? That is, does $\int D\phi \exp(-S)$ converge, or do we need to ...
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Partition Function and Expected value of dipole moment
I have a cubic lattice, and I am trying to find the partition function and the expected value of the dipole moment. I represent the dipole moment as a unit vector pointing to one the 8 corners of the ...
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Doubt in the expression of partition function of a general canonical ensemble
Suppose we have a system $S$ connected to a bath $B$. The combined system forms a microcanonical ensemble.
Suppose the energy of the combined system is $E_T$.
So, $E_S+E_B=E_T$.
The probability of ...
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AdS/CFT and entropy
I would like to calculate the entropy of a CFT on a circle, and I know the partition function of the disk. Can I use AdS/CFT:
$$ Z_{CFT} = Z_{\text{bulk}}$$
To deduce :
$$S_{CFT} = (1-\beta \partial_\...
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Problem with different expressions of functional determinant
This question is a follow-up of my previous one, after having done some calculations. In this previous question I used a minimal example of my problem with $\det(\Delta) = \det(\partial^2+A(x))$, but ...
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Functional determinant: linking Series, Heat-Kernel and Zeta function
I would like to express a functional determinant as a series of diagrams, using the zeta function renormalization applied to the heat-kernel method, but I don't know if it's possible. Let me explain:
...
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Operator insertions vs boundary conditions in AdS/CFT
This question is motivated by AdS/CFT, but really it's just about AdS quantum gravity. Consider quantum gravity in asymptotically AdS spacetime. For simplicity, assume there are no matter fields: the ...
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When can you express the Helmholtz energy as a sum of terms?
In both excess Gibbs/Helmholtz energy models the total Gibbs/Helmholtz energy is the sum of multiple contributions. For example, in the SAFT equation of state the total Helmholtz energy is
$$
A = A^\...
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What are $\mathcal{F}_g$ in string theory?
I was reading an article and came up on $\mathcal{F}_g$. Namely, it was in the following equation, $$\psi_{top} = \exp(\sum_g \mathcal{F}_g)$$ where I believe the $g$ denotes the genus of the topology ...
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Boltzmann distribution and probability of finding the system with specific energy
For sake of simplicity assume classical discrete systems.
If we have a system ($\text{S}$) coupled to a reservoir ($\text{R}$), then a microstate of the combined (isolated with fixed energy $E$) ...
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Saddle point contributions to the gravitational path integral
In his lectures on black holes and quantum information, Tom Hartman states that the gravitational path integral can be approximated as
$$
Z(\beta) \approx \sum_{g_\text{cl}} e^{-I_E[g_\text{cl}, \phi]}...
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2 State canonical ensemble partition function where number of particles in each state is known
If we're asked to find the partition function for a two state system with energies $ E_1 $ and $ E_2 $ for $ N $ indistinguishable, independent classical particles, then it makes sense to me that the ...
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Path integral in Lattice gauge theory with fixed gauge really the same as without fixing the gauge?
In 1 the question why in lattice gauge theories with gauge group $G$, there was no need for gauge fixing to obtain finite path integrals was answered. Thus observables could be calculated as
\begin{...
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Physical interpretation of the asymptotics of partition function in string theory
I would like to understand the physical interpretation of the asymptotic expansion of a partition function. The QCD partition function with gauge group $SU(N)$ as $N$ is large has been shown by Gross ...
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Why is the free energy unitless when taking the thermodynamic limit?
Why is the (Helmhotz) free energy unitless when taking the thermodynamic limit? Given the partition function $Z$ of a (finite size) system, the free energy is given by
$F =-kT \log[Z]$, where $k$ is ...
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Derivation of the Canonical Ensemble
One of the common derivations of the canonical ensemble goes as follows:
Assume there is a system in the contact with heat reservoir which together form an isolated system. Heat can be exchanged ...
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What is a symmetry of the generating functional, and what is the significance?
I cannot find a definition for a symmetry of the generating functional in Quantum Field Theory:
$$ Z[J] = \int \mathrm d \mu \, \exp\left\lbrace i S[J] \right\rbrace \, .$$
I know it's a simple ...
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Partition function for $H=-\vec \mu \cdot \vec B$
The interaction energy between a magnetic dipole $\mu$ and a fixed magnetic field $B$ is $E(\theta)=-B \mu cos\theta $ where $\theta$ in the angle between $B$ and $\mu$.
The partition function is
$Z=\...
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Confusion with Pathria's derivation of the Canonical Ensemble
I've some problems understanding Pathria's derivation of the canonical ensemble and the probability of a system being in a certain state.
According to Pathria ( section 3.2 ), we consider an ensemble ...
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Issue with path integrals for the partition function
I was going through Kapusta and Gale "Finite temperature Field theory"
In chap 2, Eq. 2.24, they need to do the path integral
$$Z = Lim_{N-> \infty} \left (\prod_{i=1}^{N} \int_{-\infty}^{...
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Generalized partition function with angular momentum
I'm studying statistical mechanics, and I found this problem in my study material:
Suppose you have a gas consisting of N identical non-interacting atoms in a harmonic trap. Consider its Hamiltonian ...
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The Partition Function of $0$-Dimensional $\phi^{4}$ Theory
My question is related with this question. Several years ago, I posted an answer to the question, and the author of the reference removed the link permanently, now I have no clue what's going on.
In ...
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Why does expressing the Faddeev-Popov determinant as this lead to such problems?
Background
In the following, I am interested in the Schwinger function associated with the gluon propagator when one considers the Gribov no-pole condition in the partition function. Defining $\nabla^{...
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Quantum versus Classical Partition Function
I am confused as to which partition functions are classical and which are quantum. I am interested in the canonical ensemble.
Several places I have seen the "classical" partition function as:...
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Ensemble associated with a system of Photons
In the book "Fundamentals of Statistical and Thermal Physics" by F. Reif it is said that :
Photon Statistics is a special case of Bose-Einstein Statistics with no restriction on the total ...
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Should vibrational internal energy of a diatomic gas be considered when calculating below the characteristic temperature $\theta_{vib}$?
(this is a confusion raised from an exercise problem in Carter's Book:)
For a diatomic gas, (take $N_2$ for example, which has $\theta_{vib}=3352k$), at room temperature: $\theta_{rot} \ll T=298k \ll ...
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Is it OK to do this manipulation at the partition function level? (auxiliary fields in quadratic gravity)
Background
I am working with the following action in the Euclidean signature ($C^2$ is the Weyl quadratic term):
\begin{equation}
S_B = -\frac{1}{2\kappa^2}\int d^4x\sqrt{g}\left(2\Lambda_C+\zeta R-\...
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String partition function
Hey I am a bit confused because when reading physics papers I encounter the free energy as a generating function of topological invariants as integrals over certain compactified moduli spaces. For ...
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Ideal Fermi gas close to $T=0 \rm K$
I am reading about the behaviour of ideal Fermi gas close to $T=0K$ from Kardar's Statistical Mechanics.
In the paragraph which I have highlighted, we have the inyegral representation of $f_m^{-}(z)$....
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High temperature limit of ideal quantum gas
I am reading ideal quantum gas from Kardar's Statistical Mechanics.
$VII.35$ is the representation of pressure, number density and energy density in the form of $f_m^{\eta}$.
$z=e^{\beta\mu}$ where $\...
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