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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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How does a microcanonical ensemble maximize entropy?

A microcanonical ensemble is one that represents an isolated system with fixed number of particles, volume, and energy. In other words, it's an $(N,V,E)$ ensemble. If the energy is fixed, it will ...
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Do quantum effects play a role in the evaporation of water?

Imagine a thermodynamic system composed of a tumbler half-full of pure water, composed entirely of 1H and 16O. The system is isolated from extraneous influences by a perfect box impermeable to ...
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1answer
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Assumptions behind Ornstein-Zernike correlation function

Let $S(\mathbf q)$ be come correlation function in Fourier space ($\mathbf q$ = wavevector). In the study of condensed matter systems, I have often encountered the statements that a reasonable form ...
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Deriving the ideal gas law from relative entropy, instead of differential entropy, to avoid negative entropy

Deriving the ideal gas law is one of the first examples presented in introductory statistical physics. It is derived using the differential entropy. Let $\chi$ be an uncountable set. Then the ...
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What actually causes blackbody radiation? [duplicate]

I've spent a fair bit of time reading up on answers detailing exactly how blackbody radiation is created but I have seen two different explanations for how it is created. The Kinetic Energy (KE) of ...
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1answer
69 views

Difficulties deriving thermodynamic cycles from a partition function

How does one produces a thermodynamic cycle from a microscopic description? When I attempt to do so, I find that the priors of my equation of state are dependant upon the Lagrange multipliers and thus ...
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1answer
85 views

The statistical physics of a simple continuous system

Let $\mathbb{R}_{\geq 0}$ be the set of real numbers greater or equal to zero. Assume an average value $\overline{R} \in \mathbb{R}$, called the prior. Then, the probability distribution $p(r), \...
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1answer
59 views

Behavior in renormalization group flow that reaches critical point

First question. Does correlation length in renormalization group flow has to be infinite when it eventually reaches critical point? Second question. Why does renormalization group flow keep partition ...
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1answer
30 views

Equation of state for a set of one element and interpretation

Assume the set $Q:=\{1\}$. We define a function $E:Q\to \mathbb{R}$, implemented as $E(q)=q$. Assume a statistical prior defined by the average value $\overline{E}$, then the probability distribution ...
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1answer
32 views

Renormalization group flow when temperature $T < T_C$, $T_C$ being critical point temperature

Does renormalization group flow have to decrease temperature when $T<T_C$, with $T_C$ being critical point temperature? I think not, but my professor suggests something like that. Maybe I ...
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Quantum Coherence in a Two-level System in the Density Matrix Formalism

Dealing with semiclassical light-matter interaction, in particular the interaction between an electromagnetic field and a two level system using the density matrix formalism, I learned that the system ...
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62 views

How do you measure the chemical potential?

It is clear how to measure thermodynamics quantities such as temperature, pressure, energy, particle number and volume. But I have no idea how to measure the chemical potential. Could someone please ...
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28 views

What are phase transiton in different contexts?

I have come across the concept of phase transitions in various contexts. From simple phase transition between different states of matter like water to ice and so on, to phase transition in magnetic ...
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43 views

Question about the Markovian property of the velocity of a Brownian particle following Langevin equation

I'am now studying Langevin model and Fokker-Planck equation with the lecture notes by Borghini Topics in Nonequilibrium Physics (NB: PDF). On page 92, he talks about the Markovian property of the ...
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32 views

Landau Theory of fluctuations

With respect to chapter 12 in the book Statistical Physics(Part 1) by Landau and Lifshitz, I am currently stuck at the intepretation of Fluctuation theory that Landau provides. In the neighbourhood of ...
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67 views

Can this be considered an Einstein model of a solid?

My abstract definition refers to energy tanks (instead of oscillators) and positive integer energy units. My defining conditions are as follows: The energy tanks are distinguishable. The energy tanks ...
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1answer
46 views

To get short-range interaction from long-range interaction

Interactions in Condensed matter systems are almost exclusively the electromagnetic interactions which are long-range. But it often gives rise to short-range interactions in systems e.g., exchange ...
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1answer
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Does the critical dynamical exponent z of a 2D Ising model (simulated with Metropolis) vary with the temperature?

I have found in the literature that the critical dynamical exponent $z$ of an Ising model simulated with a local algorithm (such as Metropolis) is something around 2 near the critical temperature, ...
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Physical systems where behavior is different when setting parameter exactly to zero

Are there any physical systems that express a certain behavior (e.g. long range order) even when some parameter limits to zero , $\lambda \to 0$, but do not have that behavior when that term is not ...
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1answer
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Why does the classical continuous partition function blow up as $T \to 0$?

At $T = 0$, we'd expect Entropy to be zero because there's only one microstate and the $\log(1) = 0$. However, when I take the limit as $T \to 0$ in the classical canonical ensemble, it goes to ...
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39 views

What is the physical meaning of time-averaged Hamiltonian?

I saw in the literature related to the systems with periodic driving forces, people often define a "time averaged Hamiltonian" as $$H_{\text{avg}}=\frac{1}{T}\int_0^T H(t) \ dt.$$ But I do not ...
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1answer
41 views

Free energy along a reaction coordinate

I've come into this issue when trying to understand biased sampling methods, in particular, umbrella sampling, but I think the question is more general. A recurring argument is that, along a reaction ...
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0answers
34 views

RG of 2D Ising with nonzero magnetic field on triangular lattice

I am given the Ising Hamiltonian \begin{align} H = K \sum_{<ij>}S_i S_j + h \sum_i S_i, \quad K>0 \end{align} to set up a real-space block-spin RG, where the renormalized spins are ...
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Derivation of density of states for a gas with $N$ states

I am trying to find any information on the derivation of the density of states for a system with periodic boundary conditions in 3D. I know how it works with 1 particle since I have seen the ...
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2answers
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Does $PV\propto T$ apply to a photon gas?

For an ultrarelativistic ideal gas, I know that $p=\frac{u}3$; $TV^3 =$ constant; $pV\propto T$. For a photon gas, I know that the first two results apply as well. However, I am unsure if the third ...
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1answer
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Eigenvalues of the Hamiltonian

Is every eigenvalue of the Hamiltonian a form of energy? If not are there values of the Hamiltonian that do not correspond to the energy of the system?
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Sound Waves and the Boltzmann Distribution

Imagine a sound wave traveling linearly in a given direction through a monatomic ideal gas. Based on gas laws and the wave equation, we have that the wave should travel, in that given direction, at a ...
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1answer
31 views

Bond order correlation function

I am trying to compute the bond order correlation function, $g_6$. It is defined based on the bond order parameter: $$\psi_6(x_i) = \frac{1}{N_i}\sum_{i=1}^{N_i}{\exp(i6\theta_i^j)}$$ where $\theta_i^...
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1answer
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Lagrange multipliers in Maxwell-Boltzmann statistics

I'm following Wikipedia's derivation of Maxwell-Boltzmann statistics. After applying Lagrange multipliers, we arrive at this expression for energy: $${\displaystyle E={\frac {\ln W}{\beta }}-{\frac {...
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2answers
73 views

Statistical Mechanics & Dynamical Systems

As a (theoretical) physics student I've taken (advanced) undergrad courses in both statistical mechanics and dynamical systems (which was purely mathematical, treatment of nonlinear differential ...
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1answer
32 views

How is heat dissipation rate the product of force and velocity?

Let $q$ be heat dissipation to midium, $F$ be the force to a particle, and $\dot{x}$ is the velocity of it. According to the equation (8) in Seifert 2005, $\dot{q} = F \dot{x}$ holds. How does this ...
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2answers
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E. T. Jaynes' subjectivism vs measurement of distributions

In his paper, E. T. Jaynes argues that entropy is a measure of our ignorance about a system. As such, the probability distribution of states $\{p_k\}$ has to be chosen in the most unbiased way, thus ...
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What is the density profile in the context of statistical mechanics?

While doing exercises in Statistical Mechanics I came across the following definition of the Density Profile of a system of $N$ non interacting particles $$\rho(\mathbf{r}) = N\langle \delta(\mathbf{...
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1answer
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Can you integrate internal energy to get original partition function? [closed]

If I have the hamiltonian of the simple harmonic oscillator $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega ^2 x^2 $$ Then it's partition function is: $$Z = \frac{k_b T}{\hbar \omega} $$ You can get ...
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Finding parameters by principle of maximal entropies of two independent systems sharing parameters

I have $n$ observations of system $A$, and $m$ observations of system $B$. $n \ne m$ Behavior of system $A$ and $B$ are independent, given the knowledge of the parameters. However, the two systems ...
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Why is it that the multiplicity function of two subsystems reaches a maximum if the $$\frac{\partial E_{1,2}}{\partial T_{1,2}}_{N,V} > 0$$

Intuitively it's clear to me and I understand why but when I try to break it down rigorously in a way that deals with the equations and their dependencies and derivatives I'm not sure that I can. ...
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How to quantify frustration for spin models with long range interactions?

Consider the following Hamiltonian: $$ H=-\sum_{i\neq j}J_{ij}S_iS_j-\sum_i H_iS_i $$ where $S_i\in\{-1,1\}$, and the summed pair $i,j$ can be any two distinct indices (not necessary adjacent spins)....
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1answer
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Problem with finding the density of states of an $N$-body system

I am having problems solving a particular problem in my Statistical Mechanics course. We have a system that is composed of $N$ non-interacting particles each of mass $m$. The particles are bound to ...
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1answer
47 views

Hamiltonian of a quantum heat bath

I have seen the Hamiltonian for a heat bath written as: $$ H_B = \hbar \int_0^\infty \omega b(\omega)^\dagger b(\omega) d\omega $$ I was hoping to understand this equation better. This suggests that ...
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1answer
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What is the meaning of $\lim_{x\to0}\frac{\ln|-x|}{\ln|x|}$? [closed]

I am now working out some critical exponent, and I encountered this result $$\lim_{x\to0}\frac{\ln|-x|}{\ln|x|}.$$ Can I write this equals to 1? Here $x=\frac{T-T_{c}}{T_{c}}$ and $T_{c}$ is the ...
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41 views

Why particles should be indistinguishable in statistical mechanics when deriving Maxwell distribution? [duplicate]

When calculating the number of microstates in statistical mechanics, we assume that particles should be indistinguishable. What is the reason behind it?
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2answers
53 views

Change of entropy in irreversible process

When calculating entropy change for a irreversible process,do I assume a reversible path and then integrated it?
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1answer
24 views

Why is the internal energy the expected value of energies of individual particles?

In this Wikipedia page: https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics) .. the total sum of energy in an ideal gas is given as: $$\langle E \rangle = \sum_s E_s P_s $$ But ...
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1answer
60 views

Entropy production, local thermodynamic equilibrium and adiabatic process

It is said that for local thermodynamic equilibrium the local entropy production needs to be 0. Now, I am reading the following from the book by de Groot and Mazur "Non-Equilibrium Thermodynamics". ...
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0answers
62 views

Partition function in the non-interacting limit

Let's consider the partition function $$Z(\lambda)=Tr (e^{-\beta H})=Tr (e^{-\beta (H_1+\lambda H_2)})$$ for a quantum system with the Hamiltonian $H=H_1+\lambda H_2$ where $H_1$ is the free part of ...
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1answer
37 views

Minimisation of Gibbs/Helmholtz free energy and Clausius theorem

I am trying to understand why (under the relevant given conditions) the free energy (either Gibbs or Helmholtz) is minimised. The derivation I have seen in several places goes like this. Set $\delta W ...
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3answers
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Non-uniqueness of the Order Parameter and its Critical Exponent

In the theory of phase transitions, an order parameter is usually defined as some quantity which distinguishes the two phases of the system by being zero in one phase, and non-zero in the other (see e....
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Ideal gas as balls in boxes in different ensembles

I have some questions about the ideal gas in different ensembles. Often boxes with moving balls are used to explain an ensemble. Now I am confused what the difference would be in the corresponding ...
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Partition function of an asteroid gas (gravity)

Consider the classical problem (Newtonian gravity) of a large number of $N$ identical non-interacting asteroids orbiting around a big planet. I wanted to see if the problem was solvable. I wrote my ...
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Modelling liquids like water with BBGKY-hierarchy

The BBGKY hierarchy is a well-known useful possibility to derive kinetic equations for gases and Plasma. The N-particle System is reduced to few-particle Systems by Integration over many Phase space ...