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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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What is the critical temperature for a BKT transition in the 2D quantum XY model with $S=1$ (not $S=1/2$)?

What is the critical temperature for a BKT transition in the 2D quantum XY model with $S=1$ (not $S=1/2$)? For instance, the classical XY model has KTc/J = 0.898 and the quantum XY model with S=1/2 ...
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$N$ independent spins, mean value of $m_N$

We have $N$ independent spins $s_i$ which can take on values $-1$ and $1$ with probabilities $p(s_i=1)=p$, $p(s_i=-1)=q$. For the total magnetization $M$ we can get N+1 different values $-N,-N+2,...,N-...
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Why does the Wang-Landau algorithm converge?

This algorithm visits every energy state an equal amount of times, and with every visit is also multiplies the density of states by a certain factor f. So how does the density of states get bigger for ...
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Mean value vs ensemble averages?

When computing the mean magnetic moment of a system with Hamiltonian $$ \mathcal{H} = -\mu \vec{H} \sum_{i=1}^{N} \cos{\theta_i} $$ and external field $\vec{H}=H\hat{e}_z$ we first evaluate the ...
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How can we evaluate the following integral using the tricks of delta functions? [migrated]

I am trying to teach myself the statistical field theory formulation of statistical mechanics. Not part of a class, just self study in my free time. I appreciate any help here. I am starting with ...
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Question about the definition of the pressure in statistical physics

I learned, that the definition of the pressure in the grand canonical ensemble is $p=-\langle \frac{\partial H}{\partial V}\rangle=\frac{1}{\beta}(\frac{\partial}{\partial V} \ln\Xi$. First thing ...
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Ferromagnet $\leftrightarrow$ paramagnet at Curie temperature

I think it's like this: $\, m=\tanh\left(\frac{Bμ}{k_bT}\right)$. If now the temperature decreases, then $\mu$ increases, until it flattens out ($\tanh$ function). Is the a point where $m$ flats out, ...
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Help in calculating Partition function and choosing the ensemble

I've been given this situation "A surface contains $N$ identical atoms in a fixed position. Every atom can occupy one of two states with energies $E_1$ or $E_2$ and the temperature is $T$." For ...
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What is the relationship between wave function and classical distribution function?

The question is a bit weird since the wave function is quantum mechanical and the distribution function in phase space is really something classical. But I would still like to know if I take the ...
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How to find equation of state of an ideal gas from heat capacity?

I'm having a problem where I have to derive the equation of state of an ideal gas from the formula for molar heat capacity $C=C_V+ \beta V$. Is this even ideal gas? Can someone help me, or at least ...
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Negative Temperature Without Statistical Mechanics

In plenty of the qualitative thermodynamics that predates the statistical description, temperature is assumed to always be positive; many key inequalities related to the second law often involve ...
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Infinite entropy in ideal gas

Does it make sense that entropy goes to infinity as temperature goes to infinity? For an ideal monoatomic 1D gas I have an expression for entropy \begin{equation} S=Nk_{B}Ln(\frac{Ve^{\frac{5}{2}}\...
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What are the excitations in the near critical 2D-Ising model in a magnetic field?

Apparently it is well known that the 2D Ising model with $T=T_C$ in a small magnetic field has a mass gap and correlation length $\xi \sim h^{- \frac{8}{15}} $. Further, in a paper in 1989 ...
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Is the air mass density in the International Space Station (ISS) higher than that on the earth at sea level?

I have read that the atmosphere in the ISS is pressurized to a standard sea-level atmosphere and also of the same composition (ratio of nitrogen and oxygen etc.). What I am struggling to find online ...
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No Schottky anomaly in the case of the heat capacity $C_{V}(T)$ of a harmonic oscillator?

Why is there no Schottky anomaly in the heat capacity $C_{V}(T)$ of a harmonic oscillator? There are Schottky anomalies present in 2-level/N-level systems with equal spacing between levels. ...
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Schottky anomaly in the heat capacity $C_{V}(T)$ of a 2-level system

Why does the Schottky anomaly in the heat capacity ($C_{V}(T)$ against $T$) associated with a 2-level system with energy spacing $E$ appear at $k_{B}T\approx\frac{E}{2}$ (a precise calculation gives $...
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boundary conditions in QM and statistical physics

I don't understand something about boundary conditions in problem that I discuss it below. in QM we solve the particle in Potential well and we obtain that we should have $k=\frac{n*pi}L$ that $n\in{...
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How to calculate the autocorrelation function of magnetic susceptibility for the Ising model?

In the paper Wolff U. 1989. Physics Letters B. 228(3):379–82, the autocorrelation time of susceptibility, $\tau_\chi$ was calculated, but the way to do so was not clearly explained in the paper. To ...
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If the spatial wavefunctions of two identical atoms overlap, but they find themselves in different energy states, are they distinguishable or not?

Say I have two quantum particles, atoms for that matter, that are completely identical in all their physical properties except that they find themselves at different locations and have different ...
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Non-composable entropies

Being not at all a specialist of the subject my question is maybe naive, I apologize if it's the case. The question is the following: There are many different version of entropy functionals around (...
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zero specific heat for infinite temperature

Why in a two possible energies system heat capacity goes to zero as the temperature goes to infinity? I thought as T increases heat capacity approached to a constant positive number, however I'm ...
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Why don’t ALL polymers collapse even when their compact state would clearly be favoured in terms of having a lower Helmholtz free energy?

This question is based on example 8.3 in Molecular Driving Forces, Dill and Bromberg (https://books.google.ch/books?id=1gYPBAAAQBAJ&pg=PA136&lpg=PA136&dq=polymer+collapse+a+toy+model&...
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A derivation in Jaynes' paper, linking stat-mech and Shannon's entropy

I have been going through E. T. Jaynes' 1957 paper, Information Theory and Statistical Mechanics. There is a step in his derivations, which has been giving me headaches for the past day; would ...
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Higgs Mechanism in Landau-Ginzburg approach

I'm experiencing some troubles with one of the exercises in Kardars book on Statical physics of fields (problem 5 Ch3) or see https://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-...
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A paradox between quantum and statistical mechanics? [duplicate]

The Boltzmann distribution function tells us what is the probability of a given particle with a given energy to be at a certain state. Now, this is in contrast with the state function the Schrodinger ...
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Observation of Bose-Einstein Condensation

Why it took us so long to see a Bose-Einstein Condensation? Was it because we didn't have the experimental setup or something else?
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Difference between fermions and bosons in Statistical Mechanics

I am an undergraduate student in Physics and Mathematics. I am now preaparing for my final exam in Statistical Mechanics and I would like some help in a particular point. So here it goes: In the ...
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1answer
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Modern uses of classical statistical mechanics? [closed]

Most of the cases when I see applications of statistical mechanics is when Fermi-Dirac or Bose-Einstein statistic are used in condensed matter or the equilibrium equation of neutron stars. Besides ...
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Proof of continuous Shannon entropy

In Jaynes' Probability Theory: The Logic of Science, there is a quick derivation of continuous entropy in chapter 12. He does so by taking the discrete definition of entropy and combines it with ...
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2answers
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Does superheated water have the same temperature as boiling water at room pressure?

According to David Tong’s notes on statistical physics on pages 139 to 140, a superheated liquid lies on an isotherm between the spinodial curve and the coexistence curve. Hence, does this mean that ...
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How can entropy always be increasing, while each microstate (among which low-entropy micro states) is equally likely to occur?

The reasoning in my statistical physics book is as follows: $\cdot$ The basic postulate of Statistical Mechanics is that for a system with a fixed energy, particle number and volume, each microstate ...
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probability of emission vs absorption in thermal radiation

I am trying to make sense of the thermal radiation emitted by a gas. The radiative transfer equation is $$\frac{dI}{dx} = \epsilon - \kappa I,$$ where I is the intensity, $\epsilon$ is the ...
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How to deduce the formula of the Correlation Length on a periodic lattice?

Sometimes in Monte Carlo simulations we need to compute the correlation length, but this is a hard task without a formula. However, for instance, in an periodic cubic lattice of $L^3$ spins, some ...
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Is it possible to implement the Invaded Cluster Algorithm on a network of Ising spins rather than a lattice?

My main concern is how to know if a cluster has percolated the network. For a periodic square lattice it is easy to determine if the cluster percolates by looking at the size of the cluster (as given ...
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Least-square fitting to data (sine function): what is the error of the derived fit parameters?

I have a set of data. I want to fit it to a sine function of the form : \begin{equation} f(x)=A sin(\omega x+B)+C \end{equation} I use the least-square method to find the appropriate fit-parameters ...
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1answer
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Partial derivatives in Boltzmann equation

In Reif's Fundamentals of Statistical and Thermal Physics, at page 499, chapter 13, he describes the Bolztmann equation in the absence of collisions and the distribution function $f(\textbf{r},\textbf{...
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Fluctuation-Dissipation Theorem [closed]

In statistical mechanics of equilibrium we say that the response is a linear function of the stimulus R(x)= cost* (1-x). How can I get it?
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1answer
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Ginzburg Criterion (Ising model)

In my statistical field theory class, we were told that we want the magnetization fluctuations in the Ising model to be smaller than their background. Specifically this was written as $$\langle\phi^2\...
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Solution of Fokker-Planck equation with constant factors by reducing it to heat equation

I am trying to solve the following Fokker-Planck equation: $$\frac{\partial}{\partial t} P(x,t|x_0) = \left( \frac{dV(x)}{dx} \frac{\partial}{\partial x} + D \frac{\partial^2}{\partial x^2} \right)...
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Electrons residing in an orbit with energy lower than the ground state energy [duplicate]

Is it possible for an electron to reside in an energy level lower than that of the ground state? What happens to the electrons when an atom is brought down to 0K , do they come closer? What happens to ...
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1answer
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Average magnetisation in the Ising Model

The Ising Model has energy given by $$ E=-B \sum_{i} s_{i}-J \sum_{\langle i, j\rangle} s_{i} s_{j} $$ where $\langle i, j\rangle$ indicates that the second sum is over each pair of nearest ...
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Energy level below ground state

Can an electron occupy an energy level lower than its ground state? Do electrons come closer to each other at 0K temperature?
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1answer
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Is it there any relation between an action and entropy?

I've found papers that seem to suggest that these concepts are the same, like this one: https://arxiv.org/abs/1005.3854 But I've found answers in Physics Stack Exchange that say that both are ...
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System of particles in classical mechanics and classical statistical mechanics

$\bullet$ Both classical mechanics and classical statistical mechanics can describe the properties of a system of classical particles. $\bullet$ In classical statistical mechanics, we assume that we ...
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456 views

Is evaporation a kind of phase transition?

When liquid is heat up to a critical temperature $T_{c}$, it starts boiling and converting to gas. In statistical mechanics, we learn that it is a phase transition. We studied all the properties near ...
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Question about notation of a Jimbo's paper

I am reading Jimbo's Introduction to Yang-Baxter Equations. And I am confused by the notation he used in the definition: Here he uses $u\in C$ without previously mentioning what is $C$. I guess ...
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Value of $\beta$ in Boltzmann statistics when degeneracy of quantum states is taken into account

The relationship between entropy $S$, the total number of particles $N$, the total energy $U(\beta)$, the partition function $Z(\beta)$ and a yet to be defined constant $\beta$ is: $$S(\beta)=k_BN \...
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1answer
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Why doesn't the equipartition theorem force the rotational energy of a monoatomic molecule to be significant?

I'm reviewing the Feynman Lectures and in them he states that the rotational kinetic energy of a monoatomic molecule in a gas is insignificant due to a small moment of inertia. But wouldn't the ...
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Why doesn't Boltzmann Approximation obey Pauli's exclusion principle?

The difference between Fermi–Dirac function and Boltzmann approximation is that Fermi–Dirac function considers Pauli's exclusion principle but Boltzmann approximation doesn't. Why?
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Degrees of freedom in M-B distributions used for solving Poisson equation

I am having some trouble to understand the Maxwell-Boltzmann probability functions or more precisely how to use it in the Poisson equation. The problem is the following: consider the case of an ion ...