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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Pressure of a gas on the inside walls of a cylinder canonical ensemble
First post on here. I need some clarification on this topic. I have an ideal gas rotating with angular velocity $\omega$ in a cylinder of radius $R$ and lenght $L$. Working with the canonical ensemble,...
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Interpretation of the ideal gas single particle partition function
When considering an ideal gas composed of identical particles, in the semi-classical case, one can find that the single particle partition function is equal to:
$$Z(1,T,V) = \frac{V}{\Lambda^3},$$
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Product of two strongly interacting canonical and grand-canonical ensembles
I am running hybrid MC-MD (Monte-Carlo and Molecular Dynamics) simulations. To put it simply, a nanocluster which atoms belong to a subset $A$ is sampling the canonical NVT ensemble via Newton's laws (...
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Computational complexity of approximating partition functions
I would like to understand the computational complexity of approximating the partition function of 2D Ising model with complex external magnetic field and complex couplings for the following cases:
...
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Integration bounds for the canonical ensemble's partition function
As far as I understand, when calculating the partition function we are summing over all of phase space.
So, for ideal gas for example we have
$$H =\sum\limits_i\frac{p^2_i}{2m}$$
and so
$$Z=\int\...
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Why do we only care for (and count) energy eigenstates and not superpositions in quantum statistical mechanics?
Consider one quantum particle in a 3D box in the microcanonical ensemble. It's energy eigenvalues are:
\begin{equation*}
\varepsilon = \frac{h^2}{8mL^2}
(n_x^2+n_y^2+n_z^2)
\end{...
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How to turn a sum into an integral?
In, An Introduction to Thermal Physics, page 235, Schroder wants to evaluate the partition function
$$Z_{tot}=\sum_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}$$
in the limit that $kT\gg\epsilon$, thus he ...
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Path integral at large time
From the path integral of a QFT:
$$Z=\int D\phi e^{-S[\phi]}$$
What is a nice argument to say that when we study the theory at large time $T$, this behaves as:
$$ Z \to e^{-TE_0} $$
where $E_0$ is the ...
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Physical observables in the XY/sine-Gordon duality
My question is, during the duality map, real physical quantities seem to acquire a prefactor of $i$ and become purely imaginary. And I feel uncomfortable.
Take bosonic current for example. Consider ...
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physical interpratation of partition function in Quantym field theory
Partition function in Statistical mechanics is given by
$$ Z = \sum_ne^{-\beta E_n} $$
For QFT, it is defined in terms of a path integral:
$$ Z = \int D\phi e^{-S[\phi]} $$
How can we see the relation ...
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1D Lattice with site dependent magnetic field
An external magnetic field is on a 1D lattice with N sites where each site has a magnetic moment, which can rotate freely. The magnetic field at the $j^{th}$ site is,
$$\mathbf{B}_j = B_0 \cos\left(\...
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Hubbard-Stratonovich (HS) transform (or similar) for higher order-interactions
I have a question about a generalization of the Hubbard-Stratonovich (HS) transformation to decouple two-body interactions. When dealing with Hamiltonians of the kind
\begin{equation}
H = -\sum_{a}...
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The partition function is given by taking the trace of a density matrix. How do I convert this sum to an integral with correct factors?
I am learning Quantum Statistical Mechanics from Kardar's Book on Statistical Physics of Particles, in that he does the following for the partition function of a 3D gas:
$Z_1 = tr(\rho) = \sum_k exp\...
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$q$-potential in grand canonical ensemble
I was recently going through R.K. Pathria's Grand canonical ensemble chapter and found the following section confusing under physical significance of statistical quantities. The author mentions;
To ...
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What exactly is the translational partition function for a single particle system?
I am recently reading Daniel V. Schroeder's book on thermal physics, and I am having trouble with the translational partition function for a single particle. Particularly since he defines the ...
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How many Lagrangians can a QFT have?
I just stumbled across a presentation by Tachikawa about "What is Quantum Field Theory". He has an interesting perspective that we should think of (at least a subset of) quantum field ...
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Different definitions of resolvent in matrix model
When I study the matrix models, I get confused of different definations of resolvent. After we define the partition function as
$$Z=\int[dM]e^{-NTrV(M)},$$
where $V(M)$ is a matrix valued function of $...
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Degeneracy of states in an ensemble of $N$ harmonic oscillators in $d$ dimensions
In the canonical ensemble consisting of $N$ independent harmonic oscillators in $d$ spatial dimensions one has to evaluate sums like
$$ \sum_n \ldots e^{-\beta E_n} = \sum_E g_E \ldots e^{-\beta E} $$
...
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Deriving partition function for various ensembles
I'm wondering the derivation of following partition functions corresponding to various ensembles:
$$
\begin{aligned}
\Xi(V, T, \mu) & =\sum_N Q(N, V, T) e^{\beta \mu N} \\
\Delta(p, T, N) & =\...
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Partition function of $N$ distinguishable particles
Suppose there are $N$ distuinguishable particles. Each particle can have energy either $\epsilon_1$ or $\epsilon_2 $. $ n_1$ particles have energy $\epsilon_1 $and $n_2$ particles have $\epsilon_2$ i....
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Distinguishability in Maxwell-Boltzmann statistics
Per the Wikipedia page on Maxwell-Boltzmann statistics, the mean occupation number describes the average number of particles in the i-th single-particle state for distinguishable particles.
To be ...
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The definition on vacuum-vacuum amplitude with current in chapter of External Field Method of Weinberg's QFT
I'm reading Vol. 2 of Weinberg's QFT. As what I learnt from both P&S and Weinberg, the generating function is defined as
$$
Z[J] = \int \mathcal{D}\phi \exp(iS_{\text{F}}[\phi] + i\int d^4x\phi(x) ...
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Calculation of canonical partition function for fermion system with degenerate energy levels
I'm having trouble in visualising the generalized version of the question asked here.
We have a system with levels whose energies are $0, \epsilon, 2 \epsilon, ..., n\epsilon$, and the number of ...
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The definition of the path integral
I still have big conceptual questions about the path integral.
According to (24.6) of the book "QFT for the gifted amateur" from Lancaster & Blundell the path integral is equal to
$$Z =\...
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What are exactly the loop correction to the potential? [duplicate]
I am struggling with the concept of quantum effective action, but first recall the definition : given a Wilsonian effective action $W[J]$ of our theory, the quantum effective action is just
$$\Gamma[\...
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The partition function of a particle in a magnetic field diverges. Why?
Using the symetric gauge $\mathbf{A} = \tfrac{B}{2} (-y, x, 0)$, the stationary states wave functions of a quantum particle in a constant and homogeneous magnetic field are
$$\tag{1}
\psi_{n m}(r, \...
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Is there any meaning or statistical distribution associated with the Jacobi's $\theta$ functions?
The Jacobi's theta functions
$$\theta_1(0,\tau )=0$$
$$\theta_2(\tau)
=\sum_{n\in \mathbb{Z}} q^{(n+\frac{1}{2})^2 /2 }$$
$$\theta_3(\tau)
=\sum_{n\in\mathbb{Z}} q^{n^2/2}$$
$$\theta_4(\tau)
=\sum_{n\...
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Yang-Lee circle theorem and the distribution of zeros
I am reading a note on Yang-Lee's zeros.
The note first focuses on the grand canonical partition function $\Xi(T, z)$, where $z$ is the complex fugacity conjugate to the number of particles. If the ...
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Making sense of stationary phase method for the path integral
I am trying to understand this paper/set of notes. I have already seen the following related question: Does the stationary phase approximation equal the tree-level term? but had some trouble following ...
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Partition function of Hydrogen atoms problem
I know there are several questions asking this problem, but I found this problem has not been solved yet to me. I will repeat the problem and state my view.
Consider the statistical mechanics of a ...
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Interpreting generating functional as sum of all diagrams
The generating functional is defined as:
$$Z[J] = \int \mathcal{D}[\phi] \exp\Big[\frac{i}{\hbar}\int d^4x [\mathcal{L} + J(x)\phi(x)]\Big].$$
I know this object is used as a tool to generate ...
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Is average total energy of two objects is the sum of their individual average energies?
In the context of Boltzmann's distribution, Schroeder states that an average is defined as
$$\bar{x}=\frac1Z\sum_sx(s)e^{-\beta E(s)}$$
Where $\beta=1/kT$ and E(s) is the energy corresponding to the ...
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Homogeneity Restrictions on the Distribution of states in Thermodynamic Systems
The expected energy in the canonical ensemble is given by
\begin{equation}
\begin{split}
\langle E \rangle &= \frac{\displaystyle\sum_{i=1} E_i e^{-\beta E_i}}{\displaystyle\sum_{i=1} e^{-\beta ...
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Examples of Path integral $\neq$ Partition function?
Are there any systems we know of whose partition function is not simply Wick rotation of the path integral? Does anyone know of any examples?
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Partition function with Hamiltonian depending on parameter
Given a Hamiltonain $H(p,q)$, I know that the classical partition function for a single particle is given by an integral over the phase space
$$
Z_1 = \frac{1}{h^3} \int e^{-\beta H(p,q)} d^3pd^3q
$$
...
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Partition function for a $SO(3, 1)$ invariant "Hamiltonian"
Suppose, I look at the $SO(3, 1)$ generalization of $H = \frac{p^2}{2m}$, i.e. $$H = \lambda P^{\mu}P_{\mu}$$ where $P^{\mu}P_{\mu}$ is a $SO(3, 1)$ invariant object and $\lambda$ is some dimensionful ...
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Probability and Single Particle Partition function
I have a fundamental question in statistical mechanics that I can't wrap my head around.
When we speak about the probability of a SYSTEM being in macrostate with energy E we calculate this by:
$$ P(E) ...
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Why non-extensive terms cancel out in the low temperature expansion of 2d Ising model?
I'm currently reading David Tong's lecture on Statistical Physics, and I cannot understand the logic in the following paragraph about the low temperature expansion of 2d Ising model.
It can be ...
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Partition function for independent particles
I am trying to understand Section 3.8.3, "Independent particles", of Piers Coleman's Introduction to Many-Body Physics (self-study, mathematics background). He considers "a system of ...
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Partition function for a classical two-particle oscillator: Infinite limits?
Previous page: What is the partition function of a classical harmonic oscillator?
Whenever I see the partition function of a classical two-particle oscillator,
$$Z(\beta) \, = \, \int dx \int dp ~ e^{-...
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Grand canonical partition function for Bose-Einstein statistic and Maxwell-Boltzmann statistics for indistinguishable particles
If we have N non-interacting particles and N assumes the values 1 and 2.They can be found in three energy levels $\epsilon_{l}=\epsilon*l$ with l = 0, 1, 2. I tried to write the function as Z(µ, T)=$\...
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$(\mathcal{S}\mathcal{T})^3=\mathcal{S}^2=+1$ mistake in CFT big yellow book?
In Conformal Field Theory Philippe by Di Francesco, Pierre Mathieu David Sénéchal
Sec 10.l. Conformal Field Theory on the Torus
eq.10.9 says the modular transformation $\mathcal{T}$ and $\mathcal{S}$ ...
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Should I partial trace the hamiltonian or partition function for a reduced system?
Suppose I have a quantum spin model, let's say e.g. the quantum transverse field model with hamiltonian $H$, on some lattice of particles, with partition function $\text{Tr}(e^{- \beta H})$ and I do ...
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Are partition functions invariant under Bogoliubov transformations?
Consider a Hamiltonian $H(a_i, a^{\dagger}_i)$ as a function of some ladder operators $a_i, a^{\dagger}_i$. Now, consider a partition function $H(a'_i, a'^{\dagger}_i)$ where $a', a'^{\dagger}$ are ...
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Constrained partition function(s) of a multi cluster system
Can a system have multiple partition functions?
I am trying to find the partition function of two cluster system with specific constraints. For the sake of brevity, I am consider a simple case. The ...
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Fixing total particle number for Canonical Ensemble vs Grand canonical ensemble
For a (non-interacting) gas of bosons (any gas for that matter), the total particle number has to be a finite value and in the canonical description this is ensured by writing a constrained sum for ...
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Partition function for fractional Brownian motion with $H < 1/2$
Recently I was interested in computing the logarithmic derivarivative $Z'(H)/Z(H)$ of the following partition function:
$$
Z(H) = \int e^{-S_H(x)} \mathcal{D} x,
\quad \text{where} \quad
S_H(x) = A(H) ...
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What is the gravitational path integral computing?
What is the gravitational path integral (which roughly goes like $\int [dg]e^{iS_{\text{EH}}[g]}$) computing?
Usually, path integrals arise from transition amplitudes such as these: $\lim_{T\to\infty}\...
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What does the matrix mean in matrix models?
I'm learning what a matrix model means, for example, in Yang–Mills matrix models, IKKT matrix model and BFSS matrix model. I have consulted a large amount of information but still not sure what the ...
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Computation of Density for an Ideal Gas
I'd like to get some help deriving the following result:
$$ \langle \rho(\mathbf{q}) \rangle = \frac{N}{V}$$
where
$$\rho(\mathbf{q}) = \sum_{i}^{N}\delta(\mathbf{q}_i-\mathbf{q}) $$
and $\mathbf{v} = ...
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