# Tagged Questions

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

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### Is the principle of indifference enough to derive the microcanonical ensemble?

The microcanonical ensemble is usual motivated solely by the principle of indifference. Textbooks usually say something along the lines of "If the only thing we know about a system is its total ...
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### Gradient effects in continuum mechanics

What I have learned is that inhomogenous materials (materials with different material properties over space and time) can be treated by the homogenization technique (https://en.wikipedia.org/wiki/...
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### Conservation of energy and realm of possibility

The law of conservation of energy states that energy cannot be created or destroyed. Based on this principle, you can safely conclude that any effect resulting from a cause must somehow keep all ...
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### Hamiltonian or free energy corresponding to 2+1D Kuramoto-Sivashinsky model

I am trying to understand if the deterministic 2+1D Kuramoto-Sivashinsky equation $$\partial_t h = -\nu \nabla^2 h - K \nabla^4 h + \frac{\lambda}{2} (\nabla h)^2,$$ where $\nu$, $K$, $\lambda$ ...
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### Intuition on Gibbs measures

I am (roughly) aware of the way Gibbs measures are used to solve physical systems (e.g. the Ising model). We can basically boil it down to pinpointing a Hamiltonian. My question is, consider a ...
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### Why don't we observe spontaneous symmetry restoration in nature?

Why do we always observe spontaneous symmetry breaking in nature and not restoration? Does there exist some argument with the 2nd law of thermodynamics and the entropy of the universe increasing? If ...
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### Statistical mechanics - average particle energy, average kinetic energy

I'm looking at derivations for average particle energy giving $E=kT$: http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/bolapp.html And average particle kinetic energy giving $K_E=\dfrac{3}{2}kT$: ...
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### Deriving the correlation function of a system interacting with a bath of harmonic oscillators

I'm working on the book Quantum Effects in Biology by Mohesni et all. My question is however not biology related, it is about a section on quantum master equations in the weak system-bath coupling ...
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### Interpreting the Fourier transform of a Gibbs measure

Recall that a Gibbs measure gives a probability distribution on states $x$ of the form $$p_X(x) = \frac{1}{Z(\beta)}\exp(-\beta E(x))$$ As I understand, the function $E$ is interpreted as the ...
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### What does 'fully excited' actually mean?

In statistical mechanics you often hear the phrases such as 'when the degrees of freedom are fully excited then....'. An example would be the validity of the equipartition theorem. But what is the ...
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### Inverting density in favour of fugacity

In these notes on pages 80 and 81 the following step was used The density in terms of fugacity is $$\frac{N}{V} = \frac{z}{\lambda^3}\left ( 1+ \frac{z}{2 \sqrt{2}} + \ldots \right )$$ and this ...
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### Is there any useful sense in which entropy fluctuates?

One of the classic distinctions between young Boltzmann and old Boltzmann was his view on entropy. Young Boltzmann had his H-theorem where a mechanical quantity H was supposed to represent entropy. ...
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### Counting the accesible microestates compatible with the macrostate conditions

Let be a system consisting of $N$ magnetic dipoles with magnetic dipole $\vec{\mu}$ in a magnetic field $\vec{B}$. I want to count the micro states accessible to the macro estate defined by $E=-\mu B$ ...
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### What happens to Boson and Fermi gases at very low temperature?

At low temperatures Fermi gases pile up to the state with the Fermi-energy having one particle in each state and Boson gases form Bose-Einstein condensates. However, the only derivations I have seen ...
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### Phase diagram binary mixture

The Gibbs Phase Rule states: F = C - P + 2. From this it is possible to construct a phase diagram. In case of a 1-component system C=1 and where P denotes the number of phases in equilibrium. In case ...
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### Violations of Onsager reciprocity?

As far as I understand it, the modern statement of Onsager reciprocity is that the linear-response transport coefficient matrix, when transposed, is equal to that of the time-reversed system (reversed ...
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### Physical reason why Prandtl number is order unity for gases?

Is there a physical reason behind the fact that for gases the thermal diffusivity is on the same order of magnitude as kinematic viscosity (and as such a Prandtl number of order unity) and if so what ...
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### Conceptual problem on Maxwell's velocity distribution law

I already read about Maxwell's velocity distribution law for gas molecule. And the expression for that distribution is following dnc=4πnA^3e^(-bc^2)c^2dc Now if we assume that the molecules have no ...
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### Statistical Mechanics: Computing a system's microstate multiplicity

I have a general question concerning how to compute the microstate multiplicity of a system, in my lecture notes, for a system of $N$ weakly coupled oscillator and $Q$ energy quantas, the multiplicity ...
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### Why does this formula for the partition function not include the multiplicity?

I am having problems understanding the formulas used for describing the partition functions and the probability distributions for canonical ensembles. In the first case I have two formulas for the ...
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### Linear thermal expansion from statistical mechanics?

I came across a question recently regarding work done by an expanding metal and the origin of the energy used for the work, and most of the responses pointed the person to look more at the enthalpy ...
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### Microscopic Definition of Heat and Work

If I am given a statistical System, then I can define state-variables like Energy, Entropy or other Observables, and then I can (at least for equilibrium states) give the Change of Energy as: \begin{...
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### Constancy of Coefficients of Additive Integrals Throughout Subsystems of a Closed System

I'm studying Landau and Lifshitz's Statistical Physics, Part 1, 3rd edition and am looking for clarification on the following statement, which appears on page 11 in the section on The Significance of ...
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### probability of striking the circular ring by gas molecules

In kinetic theory we use probabilistic case to derive pressure, no. Of molecules having speed c to c+dc or in such cases.and to derive such equations we introduce a term called "SOLID ANGLE" I come ...
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### Quantum versus classical computation of the density of states

If I consider for instance N non interacting particles in a box, I can compute the energy spectrum quantum mechanically, and thus the number of (quantum) microstates corresponding to a total energy ...
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### Mean free path in QFT

I'm trying to understand the hydrodynamic approximation of a general QFT when the large $k$ and $\omega$ DOF have been integrated out i.e that at highly enough temperature every non-trivial QFT ...
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### How the degrees of freedom for gas molecules affect energy

It seems a bit unclear to me how degrees of freedom help judge the amount of kinetics energy in the system. from the formula for the energy of molecule ε = u0 + (v/2)kT it can be inferred that the ...
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### What leads to the existence of critical temperature?

We know that $T_c$ is the temperature above which no amount of pressure could force a gas to liquefy. But why is this? Somehow I don't buy the point that the gas molecules exert too much pressure ...
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### How to write any partition function?

So I am familiar with the derivation of the partition function for a canonical and a grand canonical ensemble. I have seen definitions of the partition function for some of the quantum counterparts of ...
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### water stream cut off abruptly; why there will occur sprinkling?

Suppose that water flows with constant velocity $V$ and constant pressure $p$ through a pipe with diameter $d$. Now the pipe is suddenly cut such that the water will splash out of the pipe into the ...