Questions tagged [statistical-mechanics]

The study of large, complicated systems by means of statistics and probability theory, in order to extract average properties and to provide a connection between mechanics and thermodynamics.

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Susceptibility in the paramagnetic Ising model

In the Ising model (2D for simplicity), the magnetic susceptibility casts the form $$ \chi = \beta\left(\langle M^2\rangle - \langle M\rangle^2\right) $$ We know that the susceptibility peaks at the ...
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What is the meaning or definition of 'correlation length' in the context of Anderson localization?

I was reading a paper that talks about Anderson localization. It mentions the quantity called 'correlation length' or 'localization length' but no formal definition is given as to what it actually ...
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Grand partition function substate average number of particles

Let's say I have a grand partition function with two states $\epsilon_1$ and $\epsilon_2$: \begin{equation} Z = \sum\limits_i \exp\left[-\beta\left(\epsilon_1 n_1+\epsilon_2 n_2 - \mu n_i\right)\right]...
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What is the best equation of state for an isothermal compression of high pressure and temperature for steam

The best equation of state for isothermal compression of steam at high pressure and temperature
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Replica calculation in statistical mechanics

i am reading "Statistical physics of spin glasses and information processing an introduction - Nishimori Hidetoshi", chapter 6, page 119 on image restoration. I tried to understand what ...
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3answers
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Density of states misunderstanding in Statistical Mechanics

In the simple model of a box filled with an ideal gas, one may write the total energy as the sum of kinetic energies of all particles $$E = \sum_{i=1}^N\frac{\vec{p}_i^2}{2m}$$ and so if you construct ...
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Manipulating the Boltzmann equation for baryons

The Boltzmann equation for baryons is $$m_p\frac{\partial (n_b u_b^j)}{\partial t} + 4Hm_pn_bu_b^j + \frac{m_pn_b}{a} \frac{\partial \Psi}{\partial x^j} = F_{e\gamma}^j(\vec{x},t)\ \ \ \ \ \ \ \ eqn.(...
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1answer
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Why is symmetry not spontaneously broken in superconductors?

Take the BCS model of superconductivity. We have creation and annihilation operators for the electron, $\Psi/\Psi^\dagger$, and the order parameter for the phase transition from the metallic phase ...
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1answer
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What does the absence of these Goldstone boson interactions mean physically?

I have read that in several statistical models exhibiting spontaneous symmetry breaking, the resulting Goldstone bosons do not interact with each other via $\theta^{2n}$ terms — only via derivative ...
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Interacting fermionic models with exact analytical solution?

Is there any interacting fermionic model (i.e. with more than quadratic terms) which can be analytically diagonalised? Even the "simple looking" Hubbard model seems to lack an analytical ...
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Integrating out the collision term for the Boltzmann equation for photons

I am reading the book "Modern Cosmology" by Dodelson and Schmidt. Considering a Compton scattering process $$e^-(\vec{q}) + \gamma(\vec{p}) \leftrightarrow e^-(\vec{q'}) + \gamma(\vec{p'})$$ ...
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Can two or more bosons concretely exist at the same exact point in space at the same time?

Is it just the probability of finding the 2 particles in the same volume is the same or is it that they can really exist concretely as each other in the same point in time. Also related is, can two ...
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Nr. of microstates and macrostates for a system

Let's say we have a system S (a quantum gas,either a boson or a fermion-gas), made up by many subsystems, which we will index with $i$. One subsystem is characterized by : $\bar {\epsilon_i}$ it's ...
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Nr. of micro-states for an Einstein solid

We have $\frac N 3$ atoms in a block made of some material. That means we have $\frac N 3$ 3D oscillators or N-1D oscillators. For one oscillator, its energy is : $E_n^i= \hbar \omega(n+\frac 1 2)$, n=...
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Number of microstates for an ideal gas and its relation to the Heisenberg uncertainty principle

Considering the gas as the system, the gas particles are the building parts of it. Each gas particle has a certain value of position and momentum. For a single particle this set ( $\vec {r},\vec p$) ...
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Literature on non-equilibrium statistical mechanics [duplicate]

I am a graduate student, currently planning on self-studying non-equilibrium statistical mechanics. I have tried searching for literature myself, but no book/online course that I have found seems ...
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Why does only linear motion of the molecules contribute to temperature?

In my physics class I studied that it is only the linear motion of the molecules and not the rotation of the molecules that contributes to the temperature. But why is that? I studied that temperature ...
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Reconciling phase-space and maxent derivation of Boltzmann factor in stat mech

There is a phase-space derivation of the Maxwell-Boltzmann distribution where we start by defining the number of configurations of an ideal gas as: $$\Omega (N, V, E) = e^{3N/2} \left [ \frac{V}{\...
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Ergodicity in quantum statistical mechanics

Is there an ergodicity assumption in quantum statistical mechanics ? The classical statistical mechanics derives its main results from the assumption that all the states with the same energy (and ...
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Expanding the Bose-Einstein distribution with fractional temperature perturbation

The Bose-Einstein distribution for photons is given by $$f(\textbf{x},p,\hat{p},t) = \frac{1}{e^{\frac{p}{T(t)[1+\Theta(\textbf{x},\hat{p},t)]}}-1}$$ where $p$ is the magnitude of the momentum of the ...
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Autocorrelation and variance: can the fluctuation-dissipation theorem actually be written in terms of fluctuations?

I am considering the theorem in a statistical mechanics context, but I suppose the question could be extended to other fields where it applies as well. If we have a system with property $A$ and apply ...
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Spectrum of periodically driven Floquet operator

There is a periodically driven $XX$ model with alternating field. The piecewise Hamiltonian acts as following way \begin{equation} H_1 = \sum_{i=1}^{N-1}(\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{y}_{i}\...
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What does it mean to have delta-correlated process physically?

I am reading about Langevin dynamics, and I see the following equation: $$\frac{dx}{dt} = -\frac{1}{\xi} \frac{\partial U}{\partial x} + g(t)$$ Then, they claim that the average $$\langle g(t) \rangle ...
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1answer
58 views

Integral for the configurational part of the canonical partition function (classical monatomic gas)

S. Salinas, Introduction to Statistical Physics (p. 111) computes the following integral for the configurational part of the canonical partition function of a classical monatomic gas (volume $V$, ...
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How does the distribution curve gets modified if I take some molecules out of the gaseous system following the Maxwell-Boltzmann law?

If you have a gas of molecules following Maxwell-Boltzmann distribution then plot for the distribution curve is gaussian. Now, If I take out some molecules from it then how does the distribution curve ...
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Fermi-Dirac vs. Maxwell-Boltzmann distribution in the early universe plasma

From my studies, I remember that the quantum effects relative to the bosonic or fermonic nature of the particles play a role only in the conditions of degenerate gas: when the plasma is very dense and ...
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Lack of a name for the Thermodynamic potential wrt the natural variables

From the first law of thermodynamics relation and using Legendre Transform of the Internal Energy, we can possibly define 8 Thermodynamic potentials out of which we know 5. I could not find any names ...
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How do you calculate the average photon energy for a black body?

Let's say you have a black body at a temperature T. How do you calculate what the average energy of the photons it emits is, and if the expression is too complicated, what method would you use to get ...
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Does diffusion cause the bottle to move to the left?

There is a solution of solute and water inside the bottle, placed on a smooth horizontal surface with no friction, with the density of the solute greater than the density of the water, and the ...
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2answers
101 views

Correlation function under RG flow

I got stuck in understanding how the correlation function changes under the RG flow. Consider that the correlation function of a scalar field $\phi(x)$ in $d$ dimension is that : \begin{equation} \...
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1answer
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Bose-Einstein Distribution function

Why is the following equation : $\bar n_\epsilon=\frac 1 {e^{\beta (\epsilon_p - \mu) } \pm 1}$ called a distribution function? In wikipedia the definition of a distribution function (a.k.a Cumulative ...
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Confusion regarding the average occupation number for a Boson/Fermion

Regarding the average occupation number for a Bose/Fermi gas we have: $$\bar n_\epsilon=\frac 1 {e^{\beta(\epsilon_p - \mu)} \pm 1}$$. Now the problem I am having has to do with the nomenclature of ...
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Heisenberg Spin

I am trying to find the free enthalpy of 2 spin 1/2 particles under the effect of a magnetic field. I am given the Hamilton function expression : $H = -JS_1S_2 - H(S_1 + S_2)$ As a helpful note it is ...
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Scaling dimension in statistical field theory

I got stuck in understanding the scaling dimension in statistical field theory. Currently I am reading the statistical field theory written by Prof. David Tong. In his note(p.63), it states that the ...
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4answers
94 views

Why are the intermediate states not the equilibrium states in $S-U$ plot during a spontaneous process?

We are given an isolated system which is composed of two subsystems $A$ and $B$. The fundamental equation of entropy of both the systems are given as $$S=c(NVU)^{1/3}$$ where $c$ is a constant whose ...
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Solid-Liquid phase transition with a Lennard-Jones potential

The usual Van der Waals equation of state doesn't describe the solid phase. Nonetheless, numerically the Lennard Jones potential exhibits a Solid-Liquid phase transition, here and here for example. ...
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If dark matter can't lose kinetic energy, then why is it not traveling at relativistic speeds?

I have read this question: The only way you can do this is to remove kinetic energy from the system. With normal matter this is done through electromagnetic interactions, which turn the kinetic ...
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Helmholtz free energy for a first order phase transition

I know how entropy and specific heat for constant pressure, how both of these quantities change when we are observing the first order phase transition. I want to ask if someone can illustrate (with a ...
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1answer
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Why are critical points of classical systems described by quantum conformal field theories?

So, the question is pretty much in the title: why are critical points of classical systems described by quantum conformal field theories? I get that schematically, conformal symmetry (or rather scale ...
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Reading materials on Lee-Yang zeroes for spin systems with quenched disorder

I am trying to have a deeper understanding of the Lee-Yang zeros for spin systems with quenched disorder. So far I have read Section 3.2 of Itzykson-Drouffe which covers the concept for Ising model. ...
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Computing the partition function from a Metropolis Monte Carlo sample

I must be missing something. I could not find an answer in similar posts. Suppose I have an energy $E(x)$ and have sampled many points, $\{x_1, x_2, ..., x_N\}$ through a Metropolis Monte Carlo ...
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Temporal and spatial averaging of macroscopic variables

I am reading Callen's Thermodynamics and an introduction to Thermostatics. Initially the author describes the temporal nature of macroscopic measurements as- A macroscopic observation cannot respond ...
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1answer
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Derivation of $Z = \operatorname{Tr}e^{-\beta H + \mu N}$ [closed]

I've never studied quantum statistical mechanics myself, but I've read that the partition function of a quantum system in the canonical ensemble is given by: $$Z = \operatorname{Tr}e^{-\beta H}$$ ...
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1answer
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Integral over momentum space of the distribution function

I have $$\int \frac{d^3p}{(2\pi)^3}p\frac{\partial f}{\partial p} = \int \frac{d^2\hat{p}}{(2\pi)^3}\int^\infty_0 p^2dp\ p \frac{\partial f}{\partial p} ,$$ where $f$ is the distribution function, $\...
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Canonical ensemble heat reservoir

In canonical ensemble we treat the remainder systems as heat reservoir. Ensemble is collection of systems in all possible states of a system under study. Actually they are like mental photo copies of ...
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53 views

Tsallis $q$-Gaussian and applications

Why is not $q$-Gaussian distribution merely the substitution of q exponential into the gaussian function?, i.e. substitution of equ.2 in equ.1. Where would there be three cases as below. When to use ...
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How to get marginalized Fokker-Planck equation for the time-dependent Gaussian velocity distribution?

I have come across the term "Marginalized Fokker-Planck equation! ", which I have never heard of and could not find any resource online. The equation reads as following $$ \frac{\partial}{\...
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4answers
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Feynman Lectures Vol.1 10.5 Relativistic momentum: Why does heat energy can be easily “hidden” in random motions of the atoms of a body?

In section 10.5, Feynman says that In some of these cases, heat energy for example, the energy might be said to be “hidden.” He then further explains why the heat energy can be "hidden." ...
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Which thermodynamic states are important in the classical regime?

Let me explain my question on an example. Suppose there are 3 non-interacting particle and 4 one-particle states and all possible thermodynamic states (TDS) are listed for MB, BE, FD statistics. The ...
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Chemical spontaneity is characterized by $\Delta G<0$. Why is nuclear spontaneity characterized by $Q>0$ and not $\Delta G<0$ as well?

Chemical reactions that occur at constant temperature and pressure are spontaneous if and only if the reaction reduces the systems Gibbs free energy ($\Delta G=\Delta H -T\Delta S<0$). Clearly this ...

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