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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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Density of states for 3D simple harmonic oscillator

I have the thermal partition function and the density of states for the 3D simple harmonic oscillator, which are given below $$ Z(\beta) = \frac { 1 } { \left( 2 \sinh \left( \frac { \beta \omega } { ...
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Can one formulate a fluctuation-dissipation theorem in presence of non-Gaussian noise sources?

The fluctuation dissipation theorem relates the linear response of a system to Gaussian fluctuations. The natural question that comes to my mind is the possible derivation of an analogous FDT in ...
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Fitting an Ising Model with Probabilities

Question How to use the observations to fit an Ising model? After self-studying for several days, my current guess is: $\theta_{ii} = \log[P(X_{i} = 1)]$ $\theta_{ij} = \log[P(X_{i} = 1, X_{j}=1)]$ ...
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The mean of Langevin equation

I have a very basic question regarding the mean of the Langevin equation. So we have an equation of the form: $$\dot{v}(t)=-\beta v(t)+ \xi (t)$$ Where $\xi (t)$ is a Gaussian white noise with an ...
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Convexity/Concavity of Thermodynamic Potentials and Entropy Maximization

In a brief review of thermodynamics, our lecture notes read Thermodynamic potentials are concave in their extensive variables and convex in their intensive variables. Alright, we start with $U(S,...
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Maxwell relations for quantities per mole

Consider e.g. the differential of the Helmholtz free energy: $$dA = -SdT-pdV+\mu dn, $$ where, for simplicity, we only consider one particle species. We can then derive, for example, the following ...
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What would a non-perturbative renormalization group treatment for polymers look like?

I know that one can do perturbative renormalization for the polymer excluded volume problem or the self-avoiding walk problem corresponding to n=0 component field theory. Here in Hamiltonian, we have, ...
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How to understand the two-point correlation function in momentum space?

Let's take the Ising model as an example and study the two point spin spin correlation function: $$\langle s_0 s_r\rangle = \frac{\sum_{\{s_i\}}e^{K\sum_{\langle i ,j\rangle}s_i s_j} s_0 s_r}{\sum_{\{...
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Is the second law of thermodynamics a “no-go” theorem?

As defined here, there are several no-go theorems in theoretical physics. These theorems are statements of impossibility. The second law of thermodynamics may be stated in several ways, some of which ...
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Difference classical and statistical thermodynamics

What should I read to link between classical thermodynamics or engineerig thermodynamics and statistical mechanics?
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Total energy of a gas in an external magnetic field

Given the grand thermodynamical potential $\Omega$, the particle density $N$, the chemical potential $\mu$, the temperature $T$ and the entropy $S$, one can compute the total energy by the well known ...
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Screening of a coulomb-like potential

Imagine that we have a potential in the form of: $U(r) \propto \frac{1}{r^n}$ in a 2D system, with the high concentration of particles interacting with the above potential. How do you find the ...
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What mechanism at the microscopic level determines whether a system heats up or not?

When placed in an ordinary or a microwave oven, a beaker of water heats up except during boiling (i.e., a phase change involving latent heat). Now, suppose a system absorbs energy in such a way that ...
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Gauss Bell Curve - Non negative numbers - Will curve shape from trigonometry be the same as from empirical data? [migrated]

Im Johan, new to physics stack exchange, second post. How are you doing:) Would you help me with this Gauss Bell Curve question please? Im just looking for a general way to skew the normal ...
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Air pressure problem solving with Boltzmann factor

I am talking about the very classical thermodynamics problem, where you are given a cylinder of length $D$. The cylinder is being rotated with one of the sides as axis. You're given the pressure in ...
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Ising magnetization in metropolis

I am working on the Metropolis-Montercarlo algorithm for the square lattice ising 2D. Im running the simulations for a given lattice size, running from low temperature to high temperature, and ...
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Entropy in the Klimontovich Picture of Kinetic Theory

In the Klimontovich picture of kinetic theory, a classical $N$-body system is fully specified by the phase-space distribution $$ \rho(\vec X,t) = \sum_{i=1}^N\delta[\vec X-\vec X_i(t)] , $$ where $\...
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Rigorous derivation of the mean free path in a gas

Can anyone supply me with a derivation of the mean free path, of particles in a Maxwell Boltzmann Gas? Cited in various literature is the formula, \begin{align} \begin{split} \ell&=\frac{1}{\...
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Probability distribution of two one dimensional relativistic particles

I'm trying to solve the following problem (in preparation for my exam). Consider two one-dimensional relativistic ideal-gas particles with masses confines to a one dimensional box of length $L$. ...
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Intuitive understanding of the derivation of the Rayleigh-Jeans law

I know the Rayleigh-Jeans law and how the formula predicts UV catastrophe. Without getting into the exact derivation, I am trying to get some intuitive understanding of it by using some of the broad ...
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Does Avogadro's law only apply to ideal gases?

I know the ideal gas law may change a little when for real gases or diatomic ones does Avogadro's law change too or does it stay the same?
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What does the term $e^{-h\nu /kT}$ in Boltzmann distribution function mean and what roles does it play? [duplicate]

What is the physical meaning of $e^{-h\nu /kT}$ in the Boltzmann distribution function. I am aware that $h\nu$ represents energy of a photon and $kT$ is the thermal energy available to the system. I ...
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Understanding entropy, information, and randomness

In a statistical mechanics book, it is stated that "randomness and information are essentially the same thing," which results from the fact that a random process requires high information. More ...
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Density of states of bosons in function of momentum with energy $\epsilon = cp$

I am working out an average number N of bosons of spin $S = 0$ connected to a two-dimensional domain with surface A. The gas is ultrarelativistic with a single particle energy $\epsilon = cp$. The ...
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When does the Bose-Einstein distribution function reduce to the Maxwell-Boltzmann distribution?

When does Bose-Einstein distribution function reduce to Maxwell-Boltzmann distribution function in statistical mechanics?
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Getting the density of states for photons

I know that the density of states $g(\epsilon)d\epsilon$ is the number of states in the energy range $[\epsilon, \epsilon + d\epsilon]$. I considered a system of non-interacting free photons in 3 ...
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How does $dS = dQ/T$ define a state function $S$?

In a thermodynamic process, entropy change from a state $a$ to $b$ is defined as $$ \int_{a}^{b}\frac{\delta Q_{rev}}{T} $$ But, there can be infinitely many reversible paths to reach $b$ from $a$. ...
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Why is the partition function an integral over momentum and position?

I am learning statistical mechanics through the series of online lectures from Prof Leonard Susskind, and the partition function derived is $$Z = \sum e^{-\beta E_i} .$$ I understand this to be ...
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Volume fluctuations of a box of volume V in statistical physics makes no sense

We are all familiar with the typical example of statistical physics; $N$ particle of gas in a box of average volume $\overline{V}$ and average energy $\overline{E}$. The equation of state of of the ...
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286 views

Maximum entropy at equilibrium for closed system: Local maximum or global maximum?

For a closed system at equilibrium the entropy is maximum. Is this a local maximum or is it a global maximum? I am an undergraduate physics student and it seems that the possibility of entropy ...
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1answer
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Definition of equilibrium for thermodynamics and statistical mechanics

Is the definition of equilibrium for thermodynamics and statistical mechanics the same? From my understanding, a system is in thermodynamic equilibrium if its macroscopic variables are not changing. ...
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Question about normalizing fluid quantities (plasma physics)

I'm having trouble understanding the way the plasma fluid equations are normalized in Ch. 4 of Fitzpatrick's "Plasma Physics: An Introduction." Here's a a link to the page in question. Equations 4.147-...
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Exact solution for non-linear Fokker-Planck equation

I'm searching for exact (analytical) results for FP equation in 2 variables (such as $x$ and $p$ in 1D) with a steady state. Kramer's like (with force due to confining potential, such as harmonic ...
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Ising model and duality

I'm studying the Ising model in 2 dimensions in an approximative way. Now my professor has written this formula that links the dual space and the "normal" space: $$\sinh(2 K) \sinh(2 K^*) = 1 $$ Do ...
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Diatomic Partition Function

Given the following Hamiltonian: $H = \frac { 1 } { 2 m } \left( \left| \mathbf { p } _ { 1 } \right| ^ { 2 } + \left| \mathbf { p } _ { 2 } \right| ^ { 2 } \right) + \frac { \kappa } { 2 } \left| \...
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How do we imagine the transition between phonons and rotons?

In the theory of superfluidity, we have both phonon and roton excitations. Phonons are long-range density fluctuations (sound waves) and rotons are short-range (atomic scale) circulations of the atoms ...
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Local equilibrium of slow time varying thermal system

I'm trying to differentiate a thermal system in local equilibrium (and slow time varying) v/s a non-equilibrium system. For a thermal system which is slowly time varying, how does one define local ...
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Algebraic Bethe Ansatz state generator problem

Given $B(\lambda)=T^0_1 (\lambda)$ the component of the monodromy matrix T that creates a state, $\lambda$ the spectral parameter and $| \Omega \rangle$ the reference ground state, In "Quantum Groups ...
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What is the physical interpretation of the action integral, without the stationary action principle?

I'm still wondering about the physical interpretation of the action integral of some mechanical system (classical theory here, to simplify things): \begin{equation}\tag{1} A = \int_{t_1}^{t_2} L(q, \, ...
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Degenerate states, Boltzmann factor and statistical mechanics

The probability of finding a particle with energy $E$ according to Maxwell-Boltzmann distribution is: $$ P(E) =\frac{1}{Z}g(E)e^{\frac{-E}{k_BT}} \qquad eq(1)$$ where g(E) is the degeneracy of ...
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Product Rule for Partition Sums $Z_N=(Z_1)^N$

For the 1D Ising model with the Hamiltonian $$H=const.-\mu h' \sum_i S_i^z$$ we can write the canonical partition sum as $$Z_N = \sum_{ \{ S_i^z \}_N } e^{-\beta \mu h \sum_i S^z_i} = \sum_{ \{ S_i^...
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Question about the autocorrelation function of the fluctuating force in the Langevin model for Brownian motion

According to the Langevin model, we have, for the motion of Brownian particles, $$\frac{dv}{dt} = -M\gamma v + \zeta(t)$$ with $\zeta(t)$ the random force acting on the particle due to fluctuations. ...
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Ising Model Error Propagation

If I have the statistical uncertainties of the ensemble average magnetisation and the average energy from a monte carlo simulation of an Ising Model, how do I find the errors on the specific heat ...
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How are the coefficients determined in the high temperature expansion of the 2D Ising model?

I have been studying the 2D Ising model lately and have been looking at high and low temperatures. But I'm having problems when trying to understand the high temperature one. The final expansion looks ...
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Statistics of 1D discrete random walks

I have already asked this question in Math.SE. Let $P(n)$ be a probability distribution on the integers. Suppose a random walker starts off at the origin and, at every positive integer time, takes a ...
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Expand the partition fct. of a simple harmonic oscillator

I come across a expansion of the partition fct. of a simple harmonic oscillator $q$ as: $$q=x^{-1}(1-\frac{x^2}{24}+...) \tag{1}$$ where $x=h\nu/kT$. It’s easy to get $$q=\frac{e^{-x/2}}{1-e^{-x}}=\...
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Why are the diagonals of the pressure tensor non-negative?

I understand that the pressure tensor is simply the momentum flux which makes sense to me (pressure is force per unit area which is momentum change per unit time per unit area). From this, a simple ...
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One-dimensional Ising Model in a three spin chain

I have a system of three aligned spins with $S=\frac{1}{2}$. There are interactions between nearest neighbors, and each spin has a magnetic moment. The Hamiltonian of the system is: $$H=J[S_z(1)S_z(2) ...
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Using probability of camera flash interval to get the probability density equation in Griffith's Quantum Mechanics book

In Griffith's QM, example 1 chapter 1, what is the intuition behind using the probability of camera flash interval to get the probability density equation in terms of "dx". Griffith says that ...
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Probability at temperature in system has energy

Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a ...