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

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MIcrocanonical and Canonical - The thermodynamic limit

Considering a two level system with energies $ 0 $ and $ \epsilon$, we write out the single particle partition function with ease to be, also N-particle partition function for non-interacting ...
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67 views

How does statistical mechanics predict that hot air rises?

Does hot air rise -- from a statistical-mechanical viewpoint Question #6329 asks whether and why hot air rises. The consensus answer is straightforward: - hot air is less dense than cold air - ...
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83 views

Why do phase transitions even exist? Why not smooth density change curves?

Why do phase transitions even exist? Why not smooth density change curve? What properties of matter, quantum or otherwise, predicts that matter will undergo phases at different pressures and ...
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104 views

Strange definition of microcanonical partition function

I always thought that the microcanonical partition function would measure the number of states that correspond to some fixed energy. Despite, I found in this paper (equation 3.4) that we integrate ...
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Is there a classical analog to quantum mechanical tunneling?

In comments to a Phys.SE question, it has been written: 'Tunneling' is perfectly real, even in classical physics. [...] For sufficiently large temperatures this can put the system above a hump in ...
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1answer
41 views

How is free energy built into a Metropolis Monte Carlo simulation of an Ising model?

In the Metropolis algorithm, the change in the energy given by the hamiltonian is compared for flipping a spin. This is not the free energy, but for systems above absolute zero you are trying to ...
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56 views

H-theorem and Boltzmann equation applied to Boltzmann distribution

Using the Boltzmann equation: $$ \frac{dH}{dt} = \int_0^{\infty} dr \int_0^{\infty} ds W(r,s)[p_r - p_s][\ln{p_r} - \ln{p_s}]$$ and assuming $p_r = e^{-\beta r}$, the equation looks like $$ ...
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36 views

How can entropic effects be prevalent at low temperatures?

I read in a book that at low temperature the hydrophobic effect (for example) is entropic but at high temperatures it is enthalpic. I thought that entropy should decrease at very low temperatures. ...
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52 views

Helmholtz free energy from a relation for entropy

The Legendre transformation defines the helmholtz free energy (at least according to my lectures) as: $F(T,V,N)=E-TS$ It also says to start with $E(S,V,N)$ and $T=\frac{\partial{E}}{\partial{S}}$ ...
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98 views

Why does the Metropolis algorithm allow changes even for ∆E > 0?

In the Metropolis Monte Carlo algorithm, why can you accept changes even for ∆E > 0 (provided that a random number is less than a given probability ratio, e.g. exp(-β∆E))?
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66 views

Definition of entropy in nonequilibrium states

Thermodynamical definition of entropy $$S(p)=-\int p\ln p~dx$$ is defined only on equilibrium system. But why can't we use it for non-equilibrium system? Is there a well-accepted definition for it?
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31 views

Spontaneity / Free Energy of Non-Isothermal Process

I'm trying to determine a lower bound for the work input necessary to make an entropy-reducing process "spontaneous" in the sense that the 2nd law is not violated. For a constant temperature and ...
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30 views

The BBGKY Hierarchy

The collision term in the Boltzmann equation can be derived from the BBGKY hierarchy. Wikipedia says: In statistical physics, the BBGKY hierarchy [...] is a set of equations describing the ...
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37 views

Statistical mechanics: What is a “microscopic realization” of a system?

What is a "microscopic realization" of a system? The context is statistical mechanics. The microscopic system consists of many atoms (too many to track individually) with an assigned probability ...
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60 views

Question about Metropolis Monte Carlo in the case of equal energies

If configuration A is equal to configuration B in a Metropolis Monte Carlo method, do you still do the attempted update?
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2answers
97 views

Statistical mechanics: Meaning of “accessible” in “accessible microstates”

What does "accessibility" mean in statistical mechanics? Is it an equivalent concept to accessibility in mathematical control theory? I'll provide an example: When two systems A and B interact on a ...
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1answer
30 views

Basics of osmosis. What about excluded volume?

I may not understand osmosis very well. Let us suppose two compartments filled with water, separated by a semi-permeable membrane. At equilibrium, both levels are equals. Let us introduce now a given ...
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29 views

How do you obtain the fluctuation spectrum of a tubular membrane?

I am reading through a paper. A tubular membrane, submitted to tension $\sigma$ acting as a Lagrange multiplier to conserve area, fluctuates around a cylindrical shape of length L and radius R. ...
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62 views

What does the behavior of the pair correlation function look like in the vicinity of the critical point?

What does the g(r) look like near the critical point? I know what the pair correlation function (radial distribution function) should look like for a solid, which has regular packing and therefore ...
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39 views

Can an Ergodic dynamical system approach equilibrium?

An ergodic dynamical system $(\Omega,\phi^t,\mu)$ is such that the time average $\bar{f}$ of every function $f\in L_1(\Omega,\mu)$ equal the space average $\langle f \rangle_\mu$, i.e. the system ...
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47 views

How to calculate the ground-state energy for the Ising model?

I'm learning about the 2D ferromagnetic Ising model in zero field and trying to verify what I know by calculating the ground-state energy for the state with all 'up' spins in a 3x3 lattice. $$H = ...
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3answers
245 views

Axioms behind entropy!

The concept of entropy is very ubiquitous, we learn about its uses starting from Information Theory (Shannon entropy) up to its basic definition in statistical mechanics in terms of number of ...
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0answers
34 views

Entropy of an oscillator in Einstein's solid

This is a homework problem and I need help with it. A solid's (Einstein's model) oscillators are in the first excited state on average. How much entropy does one oscillator have? What I've tried so ...
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54 views

Thermal fluctuations in metals

My professor said that the $k_BT$ displacement in the energy levels of the band electrons is due to the space-thermal displacement of the potential of the ion host. I think that this displacement is ...
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118 views

Who invented the perfume bottle thought experiment?

A common thought experiment used to explain the second law of thermodynamics, the "arrow of time", etc. is perfume escaping from an opened perfume bottle; the perfume is likely to diffuse into the ...
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2answers
67 views

Is a superposition of (anti)symmetric states (anti)symmetric?

Let's say we have the following wavefunction of two identical particles, $A$ and $B$: $$\frac{1}{2}[(\chi(A)\psi(B)\pm\psi(A)\chi(B))+(\phi(A)\eta(B)\pm\eta(A)\phi(B))]$$ Is this properly ...
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Average ocupancy in an ideal gas at high-temperature

In David Chandler's 'intro to statistical mechanics' he states that for an ideal gas at high-temperature $$ \langle n_j\rangle=\langle N\rangle\frac{e^{-\beta \epsilon_j}}{\sum e^{-\beta \epsilon_j}} ...
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138 views

How do you measure numerically the central charge of a system?

Let's say that you are doing some Monte-Carlo simulations of a statistical system on a lattice and you observe scale invariance, meaning that you are at a conformal point. Can you get a numerical ...
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57 views

probability of sequence for given rate constants

lets consider a copolymer, $C_{r,s}^A$ containing r number of A monomers and s number of B monomers with A at the reactive end of the polymer. The equilibrium constant for A-A, A-B, B-A, and B-B bonds ...
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57 views

Use of Boltzmann over Maxwell distribution

Why is the Boltzmann distribution used over the Maxwell distribution in many cases such as the derivation of Plancks law of thermal radiation, derivation of Einstein A and B coefficients, Langevin ...
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1answer
52 views

Grand partition function of gas of non-interacting spin-1 bosons in magnetic field

Consider a gas of non-interacting spin 1 bosons in a uniform B field, each subject to a Hamiltonian of the form: $ H(\vec{p},s_z) = \frac{p^2}{2m} - \mu_0 s_z B$ where $s_z$ can take the three ...
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61 views

Dimensionless entropy interpretation

Measuring temperature in joules instead in the artificial units of Kelvin would render entropy as a dimensionless quantity. This is quite appealing since entropy has always been quite a misterious ...
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80 views

Proof of Loss of Lorentz Invariance in Finite Temperature Quantum Field Theory

In the standard quantum field theory we always take the vacuum to be a invariant under Lorentz transformation. For simple cases, at least for free fields, is very simple to actually prove this. Now ...
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What is entropy really?

On this site, change in entropy is defined as the amount of energy dispersed divided by the absolute temperature. But I want to know: What is the definition of entropy? Here, entropy is defined as ...
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55 views

Resources on Master Equations

Presently I am reading about "Introduction to dynamical process theory and simulation" which uses the notion of Master Equations to solve Markov process. I am very new to this. Can someone provide me ...
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51 views

The Equation of State for a Degenerate Fermi gas

I have read in Chandrasekhar's paper The highly collapsed configurations of a stellar mass Appendix I the equation of a degenerate Fermi gas as follows: $$n=\frac{8\pi}{h^3}\int^{p_0}_0 p^2dp$$ and ...
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2answers
49 views

What materials are used in non thermal plasma?

While reading about non-thermal plasmas, I came across their ionization potentials(~1%), and other capabilities, such as their non Maxwellian energy distributions. At what temperatures, and pressures ...
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0answers
41 views

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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0answers
101 views

Fluctuation interaction between two uncharged spheres

I'm trying to figure out quantitatively what is the force, acting between two uncharged conducting spheres and I've got stuck. It is not a kind of homework - it is just a simple act of curiosity. I'd ...
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1answer
103 views

Entropy of a two-level system

Consider a two-level system with energies and degeneracies $\epsilon_0 = 0, g_0=1$ and $\epsilon_1 = \epsilon, g_1=4$. I can show that the temperature at which both levels are equally populated is ...
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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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57 views

Time homogeneous Markov chain for Axelrod's model

I am reading paper Axelrod's model of dissemination of culture , I am unable to understand the transition probabilities of time homogeneous Markov chain for this model. Can some one please explain it ...
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55 views

How to numerically solve a complex equation? [closed]

I want to know that if you are given a very complex equation g(x)=A(T). How could you solve for x, which is a function of variable T. To be more specific, I encounter a polylogarithmic function I need ...
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3answers
592 views

Why is Avogadro's law always true?

Why is Avogadro's law always true? How and why do equal volumes of gases at equal pressure and temperature contain equal number of molecules? I know it is a fundamental principle in chemistry but I ...
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0answers
58 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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2answers
951 views

Temperature of a neutron star

In our everyday experience termperature is due to the motion of atoms, molecules, etc. A neutron star, where protons and electrons are fused together to form neutrons, is nothing but a huge nucleus ...
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0answers
37 views

Partition function of classical quadrupole in an electric field

The partition function of a dipole in an electric field is a well-known problem, analytical solvable (nice integral, can be calculated with pen and paper), for example in the Langevin treatment of ...
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69 views

Why is there a 'loophole' in Mermin Wagner for rotations?

I'm just starting out in my mathematics career by looking at some simple stuff on broken symmetries in statistical mechanics. Since 3D is 'hard' it would be very nice to look at 2D toy models of ...
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42 views

Which function denotes the energy of thermal motion within a system?

In thermodynamics, the heat $Q$ is defined as a type of energy in transfer, and is not a state function, which function denotes the energy of thermal motion within a system? 1) $TS$, (there is a ...
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Why does nuclear matter tend to maximize pressure?

I'm reading a text about equations of state of dense nuclear matter. It is often stated that the phase with maximum pressure is preferred. Why is that?