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 information content of a single-particle system?

I'm trying to better understand how information is quantified in the case of a closed system containing a single (fundamental) particle located at position $\vec{r}$ and traveling with a velocity $\...
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Is it possible to define concepts like free energy in the diffusion process?

How does the idea of free energy (which we derive from the canonical partition function) fit in the domain of non-equilibrium processes like diffusion?
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How statistical thermodynamics actually solves these problems?

Our primary objective in thermodynamics is to explain the bulk behavior of matter. In trying to do so we may think of going with the straightforward approach of explaining the bulk behavior by ...
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Maxwell distribution solely using the Stosszahlansatz and geometric transformations?

Background To quote Wikipedia: The assumption of molecular chaos is the key ingredient that allows proceeding from the BBGKY hierarchy to Boltzmann's equation, by reducing the 2-particle distribution ...
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Example of entropy and information

I was trying to write up a basic, intuitive explanation of entropy, and I wanted to see if this example I made up makes sense. Imagine a cell membrane separating our two compartments. It can contain ...
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Isn't energy absolute according to Thermodynamics?

I was taught that Internal Energy $U$ is a relative quantity: only changes in $U$ are meaningful. It doesn't have an absolute value, since it always comes with an arbitrary constant (for example $U = ...
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Why do we need an explanation for the arrow of time?

I'm reading Sabine Hossenfelder's new book Existential Physics, where she explains that because fundamental laws are time-reversible, we need an explanation for the arrow of time. Why is that exactly? ...
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Free energy minimization, statistical vs thermodynamic

When we speak about minimization of free energy there are two approaches, one from statistical physics and one from thermodynamics: Statistical physics tells that is we consider a system in thermal ...
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Why does the Planck curve drop below the Rayleigh-Jeans curve for blackbody radiation when Planck quantized the energy?

This has been a research topic of mine for days now. I understand the Rayleigh-Jeans law and how it leads to the ultraviolet catastrophe. I have been searching for a clear, conceptual explanation of ...
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What is the meaning of a thermal equilibrium between matter and radiation?

I understand that the thermal equilibrium between two bodies means that the two bodies attain the Same temperature. Therefore,there is no flow of a thermal energy between them. However, I don't know ...
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Problem in code for Monte Carlo method for frustrated Ising model

I'm studying Monte Carlo method (it's related to my school project). The code I'm using I got from a paper by Jacques Kotze (https://arxiv.org/abs/0803.0217). In his paper, he uses the Monte Carlo ...
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Boltzmann or Gibbs entropy for the canonical ensemble?

For the microcanonical ensemble the entropy, is given by the Boltzmann entropy which equals: $$S = k_\mathrm{B} \ln(\Omega(E))$$ where $k_\mathrm{B}$ is Boltzmann's constant and $\Omega(E)$ the number ...
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What is the relationship between Diode Junction Current and the Fermi-Dirac probability function?

The equation for the Fermi-Dirac distribution function is $$f(E)=\frac{1}{e^{(E−E_F)/k_BT}+1},$$ and can predict charge neutralization. How is this applied to a diode's junction current and can this ...
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Boltzmann distribution and probability of finding the system with specific energy

For sake of simplicity assume classical discrete systems. If we have a system ($\text{S}$) coupled to a reservoir ($\text{R}$), then a microstate of the combined (isolated with fixed energy $E$) ...
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How does one show that $\beta$ is the same for different substances in thermal equilibrium?

In the section regarding quantum statistical mechanics, Griffiths uses the method of Lagrange multipliers to calculate the most probable energy configuration $(N_1,N_2,\dots)$, where $Q(N_1,N_2,\dots)$...
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Can an actual physical system have zero entropy? [duplicate]

Obviously, a pure quantum state has zero entropy. It appears that in theory, this is correct; but in practice, it is not, because there is always some mixing with the environment. So, zero entropy is ...
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Fokker-Planck equation for the Wigner function to covariance matrix

I cannot understand the derivation in Louis Garbe article (https://arxiv.org/abs/1910.00604) about how to obtain the covariance matrix equation from Fokker-Planck equation for the Wigner function in ...
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How can we define a "free energy" for a protein configuration?

In my naive understanding, a free energy (either Gibbs $G$ or Helmholtz free enrgy $F$) is a property defined for an ensemble of microstates under certrain circumstances. (NPT or NVT) But I often see ...
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Can a mass on a spring oscillate when it is in contact with a heat bath?

I am reading statistical mechanics from Concepts in Thermal Physics, the author states the following after deriving the equipartition theorem. A mass on a spring has energy $E$ which is given as the ...
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What are some examples of microscopic quantities?

Mass, volume, energy, entropy, temperature, pressure are some macroscopic quantities. Which means we can think of them even without considering the molecular nature of matter. What are some examples ...
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Non-additivity of higher order terms in the intermolecular potential

The intermolecular potential energy can be written as $$u(r_{1},...,r_{N})=\sum_{i<j}^{N} u_{2} (r_{i},r_{j}) + \sum_{i<j<k}^{N} u_{3} (r_{i},r_{j},r_{k})+...$$ where the nuclear coordinates ...
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Partition Function and Probability and Perturbation

I am going through the seminal paper by Zwansky (1954). http://ursula.chem.yale.edu/~batista/classes/vaa/zwanzig1954.pdf. Now in 1st page itself, the partition function is defined as : The ...
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Langevin Dynamics simulation at different values of friction coefficient

I am a beginner in the area of Langevin Dynamics simulation. The following equation is solved numerically: $m_i\frac{d^2r_i}{dt^2}=F_{int}-\gamma\frac{dr_i}{dt}+R(t)$ In Langevin Dynamics simulation ...
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Covariance matrix of Fokker-Planck equation [closed]

Rewrite the Lindblad equation above into a Fokker-Planck equation for the Wigner function is: \begin{equation} \frac{\partial W}{\partial t}(x,p)=-\omega_0p\frac{\partial W}{\partial x} - \omega_0(Xg^...
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Sethna 3.2.1 - What does it mean to integrate over configuration space?

I'm having trouble understanding the examples Sethna uses in this section to illustrate the microcanonical ensemble. First he talks about the probability density $\rho(Q)$ that $N$ ideal gas particles ...
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What is a quon?

I have sometimes seen the term 'quon' in quantum physics and I a not sure what does it mean. Most of the time I see it as a synonym of quasi-particle but sometimes in questions like this one: Infinite ...
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It seems that enthalpy is maximized at equilibrium

For a closed system at constant pressure $dH=TdS+VdP=TdS$. On the other side we have that $dS \ge \delta Q/T$ that is $TdS \ge \delta Q$. So we have $dH \ge \delta Q$ and if the transformation is ...
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How is differential momentum assigned in multiparticle system of QFT?

I've been following Schwartz's book on quantum field theory, and got stuck at page 59 on Section 5.1 'cross section' of the book which argues that the region of final state momenta is the product of ...
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Would Maxwell's demon having infinite memory storage break the second law of thermodynamics?

Apologies if my understanding is wrong, I am literally a child. Maxwell's demon is meant to break the second law of thermodynamics by making a disorderly system orderly (see image below). But the ...
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Determining the Number of Points in the $n$-Space

Electron gas is a collection of non-interacting electrons. If these electrons are confined to certain volume (for example, cube of metal), their behavior can be described by the wavefunction which is ...
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How to understand Landau theory?

I'm trying to understand Landau theory for the last couple of days, I've read about it in Kittel's book and also on Wikipedia but can't quite get a grasp on it. I would like to understand it in simple ...
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Weinberg's proof of Gibbs' $H$-theorem

I'm trying to understand Weinberg's explanation of a general $H$-theorem he attributes to Gibbs (Foundations of Modern Physics, p. 35), but I'm having trouble with the mathematical modeling. The goal ...
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What is meant exactly by "eigenstate ensemble average"?

I am currently reading about Eigenstate Thermalization Hypothesis (ETH) and Berry's conjecture. In the paper by Srednicki on chaos and quantum thermalization, in Eq.(3.8) he calculates the average of ...
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Interpretation of thermodynamics applied to whole universe gets me confused… [closed]

While thinking about thermodynamics, I was confused about the interpretation of its fundamental concept. If we assume the whole universe as a thermal system, and describe all energy transfer as a ...
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What is the relation between the free energy and the action? More generally, what is the relation between Thermodynamics and Lagrangian Mechanics?

My question stems from the sentence said by my professor "The action is the free energy" which I don't understand. Thinking that probably I'm missing some key concepts, I'd like to know: ...
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Probability density function of thermodynamical variables in statistical mechanics

How can we, in statistical mechanics, compute the probability density function of a thermodynamical variable? I'll make a concrete example: Let's consider a Fermi gas. Usually, in statistical ...
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2 State canonical ensemble partition function where number of particles in each state is known

If we're asked to find the partition function for a two state system with energies $ E_1 $ and $ E_2 $ for $ N $ indistinguishable, independent classical particles, then it makes sense to me that the ...
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How is free energy defined for a single state?

In thermodynamics, in the canonical ensemble, it is said that the state of the system with the lowest free energy will be the equilibrium one. However, I don't understand how we can defined the free ...
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Concept of Thermal Equilibrium in the Context of Canonical Ensemble

Canonical ensemble can be used to derive probability distribution for the internal energy of the closed system at constant volume $V$ and number of particles $N$ in thermal contact with the reservoir. ...
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What is the intuitive meaning of the typical value $e^{\left\langle \log X \right\rangle}$ of a random variable $X$?

The notion of the typical value $e^{\left\langle \log X \right\rangle}$ of a random variable $X$ comes up often in the study of disordered systems. For examples see the short paragraph above eq. (4) ...
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Fugacity in the classical limit

In my study of statistical physics I have found a rather embarrassing issue. Classical thermodynamics is usually recovered in the limit $T\to +\infty$, which corresponds to $\beta=\frac{1}{k_B T}\to 0$...
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3 votes
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Fock Space in QFT vs SFT

In quantum field theory, the Fock Space decomposition of the state space is a central part of the mathematical formalism. In statistical field theories (SFT), (edit: for example, the Ginzburg-Landau ...
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Correlation length in massive scalar field theory

What does correlation length mean in free massive scalar field theory? For fields with mass $m$, the correlation length is $1/m$, implying that the fields die off after that length. So, there will ...
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Chemical potential for BE condensate: why the normalization would (uncorrectly) give $\mu> 0$?

Consider a gas of bosons. Let $g(\epsilon)$ be the distribution of energy levels (degeneracy). Consider the following integral $$I(\beta,\mu)=\int{d\epsilon g(\epsilon}) \frac{1}{e^{\beta(\epsilon-\mu)...
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Time evolution of a composite system

can anyone give me a hand in order to calculate the time evolution of a composite system consisting of two subsystems. I tried many different ways but I was not able to obtain the right answer. The ...
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The Quantum Statistical Average of the Energy-Momentum Tensor

Here: https://arxiv.org/abs/1009.3521 and here: https://arxiv.org/abs/1410.6332 as well as elsewhere, the quantum statistical average of the energy-momentum tensor is taken to be \begin{equation} \...
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Central Limit Theorem: Can not understand how to take the logartimus of Characteristic function [migrated]

I am looking at a proof of the Central Limit Theorem and I am having some trouble understanding the following step: We have the Characteristic funtion of a distribution $w(x)$: \begin{equation} \xi(...
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4 votes
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Quantum rotator and equipartition theorem

I have this dilemma with thermodynamics I can't wrap my head around. Let's say we have a molecule of at least 3 atoms that well-defined, non-zero moments of inertia around each of its axes, $I_x, I_y, ...
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Question about the recursion relation for the 1d random walk and Pascal's rule

Reading "Elements of the Random Walk" written by Joseph Rudnick and George Gaspari, i am confused by the recursion relation they gave on page 13. the book says that For 1d discrete random ...
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Bogoliubov hamiltonians for interacting bosonic gas

In the second quantization treatment of an interacting gas of Bosons, one can derive the following Hamiltonian: $$H=\sum_p \frac{p^2}{2m}+\frac{g}{2V^2}\sum_{p_1p_2q}a_{p_1+q}^\dagger a_{p_2-q}^\...
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