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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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14 views

Transformation to get rid of distribution multiplets

Are there any well-known distribution transformations that account for doublets, triplets, higher multiplets due to time series independent peaks occasionally being too close to resolve? I feel like ...
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1answer
30 views

Quantum probability distributions via canonical traces

Given $N$ distinguishable quantum particles in the canonical ensemble, we can estimate the probability of finding one of those, labelled by $j$, in a certain position $x\in\mathbb R$ by computing \...
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Trick for calculating the thermal entropy of the SYK model in the large N limit

Reading about the SYK model I encounter a trick that should help to calculate the thermal entropy of the system. I am not able to understand what they are doing though. Particularly the part from eq....
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25 views

quantum order in a (generalized) thermal steady state

It is known that starting from an initial product state, non-integrable systems will thermalize and eventually local observables can be described by a Gibbs ensemble. It has also been argued that a ...
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36 views

Partition function for generic spin state

I am studying statistical mechanics starting with the Gibbs state and the postulate of the partition function. I learned that the partition function is a sum over all the possible states of a system ...
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86 views

Relation between entropy and information. Is entropy absolute or relative?

I always thought of entropy as absolute, i.e, many people can measure it for a system and they will all come up with the same number. However, the entropy in statistical mechanics is defined as -Klog(...
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1answer
48 views

Potential energy of an ideal gas

According to equipartition theorem, for ideal gas in thermal equilibrium, each vibrational mode will get $kT/2$ for kinetic energy and $kT/2$ for potential energy. But at the same time, we assume ...
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1answer
22 views

Spatial Correlation Function and Ensemble average

Well, I was reading the Statistical Mechanics book by Pathria, to understand the concepts of the correlation function. I want to quote some lines. Spatial correlation functions are based on n-...
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1answer
52 views

What does it mean for the grand potential $\Phi$ to be minimised for a process at constant $T,V,\mu$, when $\Phi$ is a function of $T,V,\mu$?

I've read from a few places like Kjellander, R. (2019). Statistical Mechanics of Liquids and Solutions: Intermolecular Forces, Structure and Surface Interactions Volume I. p.83. and this Physics Stack ...
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30 views

Semi-classical limit of GC partition function

In one of the problem sets for my course on statistical mechanics, there was a question asking us to consider the semi-classical limit of the partition function $\exp(-\mu / kT) \gg 1$. My question is ...
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31 views

Eigenfunctions and eigenvalues XY Ising model

In the XY Ising model we have the following Hamiltonian: $$H=-J\sum_i \cos(\theta_i-\theta_{i+1}).$$ From this I found that $\langle \theta_i| T | \theta_{i+1}\rangle = \exp(\beta J \cos(\theta_i-\...
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Eigenvalues of transfer matrix Ising model spin 1 system

I am calculating the partition function of an Ising model with spin 1 ($\sigma_i \in \{-1,0,1\}$) for $n$ sites. The following Hamiltonian has been used: $$H = -J \sum_{i=1}^{n} \sigma_i\sigma_{i+1},$...
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15 views

Metal resistivity temperature dependence in free electron model

In the context of the Drude model, I understand why the conductivity of a metal should decrease with increasing temperature (the conductivity scales linearly with scattering time $\tau$, which due to ...
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17 views

Statistical mechanics transfer of energy

I have a question related to energy transfer from a hot to a cold system. One having 3 units of anergy and other 5. Having both 3 particles/molecules. So, how do I represent the transference of energy ...
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23 views

Question about the cumulant expansion in Kardar's book

I have one question relating to cumulant expansion when learning statistical mechanics using Kardar's textbook.To calculate the partition function for intertacting system, it runs as following The ...
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29 views

Double Sommerfeld Expansion

Consider the Fermi-Dirac expansion for an arbitrary function $f(\epsilon)$: $I(f)=\int_0^\infty d\epsilon\frac{f(\epsilon)}{e^{\beta(\epsilon-\mu)}+1}$ The large $\beta$ expansion of this quantity ...
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20 views

Acceleration statistics at thermal equilibrium for a simple atomic system

Consider a case of a simple liquid, say argon or Lennard--Jones potential. What are the statistical properties of acceleration at equilibrium? For example, acceleration distribution or a correlation ...
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89 views

Is Fermi energy the same as chemical potential?

In all of the textbooks , the Fermi Energy was interpreted as the energy level below which all possible energy states are filled. However with a Fermi-Dirac type of distribution, which has asymptotes ...
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22 views

Electric dipole in a larger dipole field

I have a problem describing a smaller dipole moving in a field from a larger 'static' eletric dipole. I've derived the potential energy $V = \frac{pp'}{4\pi\epsilon_0 r^3}[3\cos\theta\cos\theta' + \...
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3answers
70 views

How can the pressure of the gas and that on the piston be different?

In the book of Kondepudi & Prigogine, Modern Theormodynamics, at page 114, it is stated that In a reversible expansion of a gas, the pressure of the gas and that on the piston are assumed to ...
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39 views

Static Structure Factor

I know that static structure factor can be calculated from molecular simulation or X-ray diffraction. I would like to ask that if there is a way I can also calculate static structure factor given a ...
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1answer
46 views

Mutual Equilibrium (Thermodynamics)

From Wikipedia, https://en.wikipedia.org/wiki/Thermodynamic_equilibrium The second paragraph states that "Systems can be in one kind of mutual equilibrium, though not in others" . Could someone give ...
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25 views

Kinetics of thermal fluctuations in liquid

I have an instinct that tells me that particles in water at temperature $T$ should be receiving and emitting "quanta" of energy of size $kT$ at a certain rate $r$. My question is: what is $r$? This ...
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1answer
55 views

Meaning and mathematical origins of “$S$ lies in a small range $\overline{E}\pm \Delta \overline{E}$” as used in statistical physics?

The fundamental relation of thermodynamics is: $$ dS=\beta d\overline{E} - pd\overline{V} $$ It is exact for infinitesimal variations of $\overline{E}$ and $\overline{V}$ and it can be integrated ...
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Relation between scaling dimension and critical exponents for harmonic peturbations in $O(N)$ Wilson-Fisher (WF) in an old paper

I am reading the paper "Harmonic perturbations of generalized Heisenberg spin systems" (D J Wallace and R K P Zia, 1975) - https://iopscience.iop.org/article/10.1088/0022-3719/8/6/014/meta . The ...
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1answer
106 views

For irreversible processes , why is $\frac{Q_1}{T_1} < \frac{Q_2}{T_2}$?

In the book of Kondepudi & Prigogine, Modern Theormodynamics, at page 97, it is stated that All real heat engines that go through a cycle in finite time must involve irreversible processes ...
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51 views

Finding the radius of the nozzle; how to get the pressure during the ejection of $O_2$?

In the book of Kondepudi & Prigogine, Modern Theormodynamics, at page 87, in question 2.18, it is asked that $$ \begin{array}{l}{\mathrm{O}_{2} \text { is flowing into a nozzle with a velocity ...
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130 views

Statistical physics is unable to prove that $TdS=d\overline{E}$

I will pose $k_B=1$. Suppose a system of statistical physics with the constraints: $$ \begin{align} 1&=\sum_{q\in\mathbb{Q}}\rho(q)\\ \overline{E}(\beta)&=\sum_{q\in\mathbb{Q}} E(q)\exp(-\...
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158 views

Can I derive the 1 dimensional Maxwell-Boltzmann Distribution using this method?

I am aware that the 1D MB Distribution is usually derived by counting the momentum states in various directions. However, I would like to know if the following method using the 3 dimensional speed ...
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2answers
61 views

Bose-Einstein condensation: Bogoliubov Approximation

I'm trying to understand the Bogoliubov approximation from "Statistical Mechanics" by Pathria and Beale. First of all they say Since $a_0^{\dagger}a_0=n_0=O(N)$ and $(a_0a_0^{\dagger}-a_0^{\dagger}...
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1answer
53 views

Sampling Maxwell-Jüttner distribution for non-unity mass and speed of light

I am trying to sample Maxwell-Jüttner distribution using the Sobol method as described in Zenitani Loading relativistic Maxwell distributions in particle simulations (2015). Equation (2) in the paper ...
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1answer
38 views

Finite Size effects in phase transition

I was reading Nigel goldenfeld's Lectures on Phase Transition and Renormalisation Group and came across the following statement: 'If there were perfect instrumental resolution, a change in the ...
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20 views

How to calcultate the interaction energy of a spin in a ferromagnetic material ? (Heinsenberg model)

In a ferromagnetic material, one cannot set $B=\mu_0 H$ because $M$ cannot be neglected. Then I don't understand why we often read that the energy of a single spin in the material is equal to (...
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1answer
51 views

How does phase transition occur in finite sized ising model?

I was simulating the square lattice Ising model via Metropolis Algorithm and found that at 0 magnetic field, there is spontaneous magnetisation below some temperature. I have used Periodic Boundary ...
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3answers
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How do we tell which part of kinetic energy gives rise to temperature?

I know that macroscopic temperature is a measure of kinetic energy of particles at very low scales (let's call it microscopic kinetic energy). But how can we derive which part of this microscopic ...
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2answers
257 views

Why isn't temperature definable for a single, classical particle? [duplicate]

I read that the relation between temperature and kinetic energy of an ideal gas is only applicable to a large number of particles so that the mean value for kinetic energy has to be considered, so the ...
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64 views

Generalised Ising models?

Are there generalised Ising models: The underslying mesh/connectivity is completely arbitrary - non rectangular, 3D...ND, complete connectivity should be possible The interaction potential is ...
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2answers
86 views

If $dQ_p = dU_p + pdV = dH_p$, then how can $dQ_p / dT = \partial H_p / \partial T$

In the book of Kondepudi & Prigogine, Modern Theormodynamics, at page 65, (under constant pressure) $$dQ_p = dU_p + pdV = dH_p,$$ where $H_p$ is the entalpy at the constant pressure $p$. ...
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1answer
62 views

Electrons with disorder & something like AdS/CFT duality

I know that consideration of electrons with disorder can be based on Feynman diagrams with disorder lines. In this approach, only non-crossing diagrams are important and give contribution to self-...
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1answer
62 views

Mixing cold with hot water - How long does it takes?

Sometimes the bath for my baby is too hot, so I mix into it some cold water. I open a cold water stream and mix the water with my hand waiting for the temperature to drop. But here is something that ...
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32 views

Heat-bath algorithm of 2D XY model

Can anyone suggest any reference where the heat-bath algorithm for the classical 2D XY model has been discussed in detail. I have found references for the 3D Heisenberg model which can be exactly ...
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1answer
49 views

The transversality axiom in Lieb's Thermodynamics paper

I'm reading The Physics and Mathematics of the Second Law of Thermodynamics and have a question about the T4 transversality axiom which is writtern on page 54. T4) Transversality. if $\Gamma$ is ...
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1answer
102 views

Does the postulate of equal a priori probability apply only to equilibrium states or to all states satisfying the constraints?

In deducing the zeroth law of thermodynamics in micro-canonical ensemble, there is a frequently-mentioned example. Suppose we put two isolated system, system 1 and 2, in contact and allowing them to ...
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27 views

What is the role of the density distribution function by Liouville equation in statistical physics?

Constant density is a solution of Liouville equation which says that total derivative of density distribution functions with respect to time is zero, and it is the distribution function in ...
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3answers
136 views

How long does it take for an electron to reach equilibrium with blackbody radiation?

While teaching a course on electrodynamics, I thought of an interesting question that I think deserves some attention. Consider an ensemble of electrons all with momentum $\hbar \mathbf{k}$ ...
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45 views

Deriving Bolzmann velocity distribution from the argument the author provides

In the book of Kondepudi, Modern Thermodynamics, at page 28, it is given that According to Boltzmann principle, the probability that a molecule will have a translational energy $E_{trans}$ is ...
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Is there any topics in thermal physics that I need know when studying spintronics?

I have a plan to enroll the graduate school and study spintronics next year So before, I'm reviewing the undergraduate physics course now and it's time to study statistical and thermal physics. But ...
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1answer
23 views

If at a given value of reduced volume and temperature, all gasses have the same reduced pressure, shouldn't have value be constant throughout?

In the book of Kondepudi, Modern Thermodynamics, at page 21, it is given that Every gas has a characteristic temperature, volume and pressure; $T_c$, $V_c$, $p_c$ which depend on molecular size ...
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1answer
97 views

Einstein solid degree of freedom

I was studying from Schroeder's thermal physics book. When it talks about Einstein solids it says that they have 2 degrees of freedom thus $U=NkT$ However, I thought when we talk about Einstein ...
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1answer
77 views

Question about using Liouville's theorem to calculate time evolution of ensemble average

With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\...