The study of large systems through coarse graining microscopic descriptions, providing a more detailed understanding of thermodynamics.

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Finding the moments of the Boltzmann/Gibbs Distribution

I am trying to compute the moments of the Boltzmann distribution using a moment generating function, by taking the Fourier transform of the distribution and then taking derivatives to find the ...
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44 views

Why does decay of correlations imply absence of order?

In a few articles I have read, a two-point correlation function $\langle g(x)g(y) \rangle$ is shown to decay with increasing distance of $x$ and $y$, and this is then taken to imply an absence of the ...
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41 views

Fluctuation spectrum of lipid bilayer membranes

I am interestend in calculating the fluctuation spectrum of a thermally fluctuating 2d membane which is only subject to a surface tension $\sigma$. ($\mathcal{H}=\sigma\int\mathrm{d}A$) Depending in ...
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40 views

Is there a reasonable lower bound for free energy per site of the 2D Ising model in the presence of an external field?

Given the standard Ising partition function: $$Z(\theta ,h) = \sum\limits_{\bf{x}} {\exp \left\{ {\theta \sum\limits_{(i,j) \in E} {{x_i}{x_j}} + h\sum\limits_{i \in V} {{x_i}} } \right\}}, $$ is ...
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77 views

Reference for stochastic processes which helps moving from a basic level to a measure theory one

I'm looking for a reference (books, notes, lectures) which helps a physicist to understand the language of measure theory in the context of stochastic processes (in particular markov chains). I've ...
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32 views

How to load Bose-Einstein Condensates into an optical lattice?

In cold atom experiments, what techniques are used to load Bose-Einstein Condensates into an optical lattice??
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92 views

Statistical mechanics of a coin toss

I'd like to ask some questions about flipping two coins related to statistical mechanics, e.g. microcanonical distribution, phase space distribution function etc... after I rephrase the coin flipping ...
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63 views

Three-body correlation function in kinetic theory

In Kinetic Theory, one studies the evolution of a system of $N$ particles interacting with each other. We use the notation $\boldsymbol{w}_{i}$ to describe the coordinates in phase-space of each ...
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94 views

Lennard-Jones induced pseudo-molecules

It can be shown that the Lennard-Jones potential - which describes the interaction between particles in non-ideal gases - gives rise to pseudo-molecules: after a triple "collision" of three ...
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62 views

Statistical Mechanics: Most probable orientation of grain particle in gas chamber?

I'm in an introductory statistical mechanics course, and we've been posed the following situation: Long-shaped dust particle (so imagine something like a grain of rice) is placed in a gas chamber (so ...
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48 views

What is the difference between thermodynamical equilibrium and statistical equilibrium?

I am trying to understand what is the different between thermodynamical equilibrium and statistical equilibrium, for example, between photons and electrons at the early universe. (I read through paper ...
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74 views

Motivation of the Heisenberg model of ferromagnetism

In the Heisenberg model of ferromagnetism the atoms are assumed to be arranged in a lattice. The $i$-th atom has a spin operator $\vec S_i$ (here $i$ belongs to the lattice). The Hamiltonian is given ...
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185 views

How to derive the two-term approximation for the Boltzmann equation?

Starting with the Boltzmann equation in terms of $f(t,\vec v,\vec x)$ or $f(t,\vec v)$ http://en.wikipedia.org/wiki/Boltzmann_equation $$\left(\frac{\partial}{\partial t} + \vec{v} \, ...
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71 views

Can classical orders coexist with quantum orders?

For example, the ground state of the antiferromagnetic(AFM) Heisenberg model $H=J\sum_{<ij>}\mathbf{S}_i \cdot \mathbf{S}_j(J>0)$ on a 2D square lattice is a Neel state, which is a classical ...
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52 views

Random orientation percolation (Grimmett model) from the viewpoint of statistical mechanics

This is a rather soft question, but I would like to know how physicists would approach a problem which seems to be hard from the mathematical prospective. The Grimmett percolation model is defined ...
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33 views

What is the lifetime of an induced magnetization in a para/diamagnetic material?

To the best of my knowledge thermal fluctuations are responsible for washing out any effective magnetization, once the external field is switched off. Since thermal fluctuations need some time to ...
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238 views

Phase transitions. Conceptual link of my intuitive notions and definition of Georgii's book in terms of probabilities

In his classic book O. H. Georgii (Gibbs Measures and Phase Transitions) in Chapter 2 p. 28 define the concept of phase transition follows. Definition A potencial $\Phi$ will be said exhibit a ...
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91 views

Phase diagram of SO(5) rotor model

It was originally a problem from Professor Eugene Demler's problem set. Consider an SO(5) rotor model: \begin{align}\mathcal{H}=\frac{1}{\chi} ...
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109 views

Derivation of impact free Boltzmann equation

When deriving the impact-free boltzmann equation ( $\frac{\partial f}{\partial t} + \vec{v} \cdot\frac{\partial f}{\partial \vec{x}} + \vec{a} \cdot \frac{\partial f}{\partial \vec{v}} = 0$) I have a ...
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62 views

error propagation and collision in ideal gas

When dealing with gas, a statistical approach is needed because For N particles, you have to solve 6N equations which cant be done analytically. To know our time step for numerical solving, you can ...
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58 views

Relevant operators in two dimensional O(n) models

The most general hamiltonian of a two dimensional $O(n)$ and $Z_2$ invariant statistical model can be written: $$ H=\int d^2 x \left[\frac{\nabla \mathbf{\phi}^2}{2} + \frac{m_0^2}{2}\mathbf{\phi}^2 ...
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63 views

What is the minimum non-integer dimension for which the XY model shows a phase transition? (if well-defined)

I know that XY statistical model for $d=2$ doesn't show a regular phase transition , while the $3d$ has, I was wondering what is the behaviour for $2< d < 3$. If it is simpler one could ...
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114 views

Ising Hamiltonian for relativistic particles

An Ising system is described by the simple Hamiltonian: $$H = \sum\limits_{i} c_{1i} x_{i} + \sum\limits_{i,j} c_{2ij} x_i x_j \,\,\,\,\,\,\,\,\,\,(1)$$ Here the $x_i$ are spins (+1 or -1 in units ...
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423 views

Pauli paramagnetism for electrons with external magnetic field

Apparently it is to be shown that for electrons under an external magnetic field, in the limit as $B\to 0 $ $$ \chi = \frac{dM}{dB} \approx \frac{n\,\mu^{*^2}}{k\,T}\,\frac{f_{1/2}(z)}{f_{3/2}(z)} $$ ...
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135 views

Deriving the “total” Bose Einstein density of states, including the condensate

Is is possible to derive the Bose-Einstein density of states containing the delta function representing the BE condensate?
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170 views

Spontaneous symmetry breaking in the quantum 1D XX model?

The ground states of the quantum 1D Ising and Heisenberg models exhibit spontaneous magnetization. Is this also true for the 1D XX model?
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97 views

Spin Glass Transitions in Random Bond Ising Model (RBIM)

In brief, is there a list of spin glass transition properties for the RBIM on different lattices? Is there any know results about the relationships between these probabilities for a graph and its ...
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114 views

Semiflexible discrete polymer chain

Suppose we have a 2D polymer model described by a set of 2D vectors {$\mathbf{t}_i$} ($i=1,2,\dots N$) of length $a$. The energy of the polymer is given by: $$ ...
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86 views

Stat mech explanation for separation of one liquid from another in gravity?

If one mixes two distinct ideal gases above the Earth's surface, one with a higher molecular mass than the other, then at equilibrium, their number density gradients will be such that at low heights, ...
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96 views

Randomly sampling a “well-mixed” solution of Brownian particles

I place $N$ Brownian particles in $V$ liters of solution, shake until I assume that the particles are "well-mixed", and sample and randomly sample an $S$ liter volume. What is the probability ...
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78 views

Factorization of fermionic scattering integral in 2d momentum rep

the scattering integrals for fermions involves both momentum ($k$) and energy ($k^2$) conservation and a nonlinear phase space factor of a distribution function $f(k)$. $$\begin{multline}I(k) = ...
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220 views

How is the “negative dispersion” derived?

I'm looking at Kopfermann H., Ladenburg R., Nature, 122, 338-339 (1928) and it appears Ladenburg in Ladenburg R., Z.Physik, 4, 451-468 (1921) was the first to discover the phenomenon of "negative ...
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17 views

When would the Gross-Pitaevskii equation break down as $a\rightarrow \infty$?

It is now common to use Feshbach resonance to tune the s-wave scattering length of a Bose-Einstein condensate. Apparently as $a\rightarrow \infty$, the GPE would break down. The reason is that it ...
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30 views

Understanding the phase separation of a chemically reactive mixture

Let's consider a mixture of solvent and components $A$ and $B$ such that $A$ can transform into $B$ with a rate $k_1$, and the reverse reaction with a rate $k_2$. $A \rightleftharpoons^{k_1}_{k_2} B ...
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20 views

Wigner function vs. backward kolmogorov

We know that a Wigner function is a quasi probability distribution and a backward Kolmogorov equation for a stochastic differential equation (say quantum Lagenvin equation) gives the probability ...
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40 views

ensembles and lagrange multipliers

In the derivation of maxwell-boltzmann distributions, the method of Lagrange multiplier is $\sum n_i = N$ $\sum n_i E_i = E$ where $N$ is the total number of particles, and $E$ is the total ...
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47 views

What excactly is a “fourier component of a density fluctuation”?

Light scattering texts say depending on the scattering angle, you are seeing a certain fourier component of a density fluctuation. This density fluctuation varies sinusoidally due to Brownian motion ...
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66 views

What algorithms can be used to compute the binodal in a ternary Flory-Huggins theory?

What are the most popular algorithms used to obtain a binodal curve for the ternary mixture (starting from Flory-Huggins theory)? I would like to obtain a plot similar to the one calculated here ...
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72 views

Ergodic Hypothesis; canonical ensemble

I'm currently studying for an exam in thermodynamics/classic statistical mechanics and 2 things came up which are confusing me. First the ergodic hypothesis states that it is equal to take the mean ...
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25 views

Methods for quantifying a network of coupled oscillators

I usually am more on the statistics part of things, so pardon my misuse of the terminology. I am simulating a network of non-pulse coupled non-linear oscillators ( I am not sure what the correct term ...
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66 views

Diamagnetism of a degenerate electron gas for weak fields

In the book "Statistical Physics, Part I ($3^{{\rm rd}}$ edition)" by Landau and Lifshitz, at $\S59$ when he treats the diamagnetic part of the magnetisation of a degenerate electron gas for weak ...
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45 views

Free energy a continuous function of temperature but may not be differentiable everywhere?

So according to my understanding, the free energy of the system should be a continuous function of temperature. This is because if the free energy is not continuous at temperature T, then at this ...
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27 views

Effusion of particles from one box to another - pressure calculation

Suppose we have a container divided into equal halves. Right half is fixed at temperature $T$, volume $\frac{V}{2}$. Initially it has pressure $P_0$, a hole of area $A$ is opened between them. I ...
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83 views

Connection between String theory and Statistical Physics

I would like to think via standard transitivity arguments that there should be a deep connection between String theory and Statistical Physics. Why? Statistical Physics $\rightarrow$ QFT 2d QFT ...
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54 views

Calculating heat capacity from the equation of state

It is known that within thermodynamics alone, given the equation of the state of a system, one cannot explicitly determine the heat capacity. What is the mathematical reason for this? Intuitively, it ...
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52 views

Derivation of Higher-order correlation functions from definition

I'm trying to understand the definition of the n-th order correlation function. My aim is to translate the math into a numerical implementation in order to compute the correlation function $g^{(n)}$ ...
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13 views

Coarse-graining on a second channel decreases mutual information?

Let $X_1,B_1,X_2,B_2$ and $Y_1,A_1,Y_2,A_2$ and $C_1$ and $C_2$ be binary random variables. Suppose: $I(X_2:B_2|C_2=0)+I(Y_2:A_2|C_2=1) \leq 1$. This can be thought of as a bound on the capacity ...
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102 views

Fluctuations in energy for macro and micro canonical ensembles

I was thinking about fluctuations in energy for a system in thermal equilibrium. I think that the Boltzmann distribution itself has an standard deviation approximately equal to the mean, as it is ...
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44 views

Fluctuation-Dissipation theorems in an infinite quantum system

So for a quantum spin chain, one can easily prove via the partition function that you have a fluctuation-dissipation type relation between the magnetic susceptibility and the variance of the ...
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108 views

Understanding the product of partition functions by making sense of the maths and the physics

I have $N$ distinguishable particles in a 1D harmonic oscillator potential with 'proper' frequency $\omega$. The particles also have internal spin-$\frac12$ degrees of freedom in a magnetic field $B$ ...