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46

The ergodic hypothesis is not part of the foundations of statistical mechanics. In fact, it only becomes relevant when you want to use statistical mechanics to make statements about time averages. Without the ergodic hypothesis statistical mechanics makes statements about ensembles, not about one particular system. To understand this answer you have to ...


32

We can only approach absolute zero asymptotically because we can't suck heat out of a system. The only way we can get heat out is to place our system in contact with something cooler and let the heat flow from hot to cold as it usually does. Since there is nothing colder than absolute zero, we can never get all the heat to flow out of a system. We can ...


28

As for references to other approaches to the foundations of Statistical Physics, you can have a look at the classical paper by Jaynes; see also, e.g., this paper (in particular section 2.3) where he discusses the irrelevance of ergodic-type hypotheses as a foundation of equilibrium statistical mechanics. Of course, Jaynes' approach also suffers from a number ...


27

The separation does not violate the 2nd law of thermodynamics, because the oil and water phases being separate is a lower energy state. The water molecules strongly interact with each other, forming hydrogen bonds. The protons of water are shared between two oxygen atoms of two different water molecules, forming a constantly changing network of molecules. ...


24

Arnold Neumaier's comment about statistical mechanics is correct, but here's how you can prove it using just thermodynamics. Let's imagine two bodies at different temperatures in contact with one another. Let's say that body 1 transfers a small amount of heat $Q$ to body 2. Body 1's entropy changes by $-Q/T_1$, and body 2's entropy changes by $Q/T_2$, so ...


23

If you have only one species of particles then working with $(\mu,p,T)$ ensemble does not make sense, as its thermodynamic potential is $0$. $$U = TS -pV + \mu N,$$ so the Legendre transformation in all of its variables (i.e. $S-T$, $V-(-p)$ and $N-\mu$) $$U[T,p,\mu] = U - TS + pV - \mu N$$ is always zero. The fact is called Gibbs-Duhem relation, i.e. $$0 ...


21

Actually, temperature is defined as $$\frac{1}{T} = \frac{\partial S}{\partial E} = \frac{k_B}{\Omega}\frac{\partial\Omega}{\partial E}$$ So in order to have zero temperature, you would need a system with either zero multiplicity, which you can't have by definition, or an infinite derivative $\partial\Omega/\partial E$ even though the multiplicity itself ...


21

First, strictly speaking a neutron star is not a nucleus since it is bound together by gravity rather than the strong force. Measuring a surface temperature for any star is deceptively simple. All that is needed is a spectrum, which gives the luminous flux (or similar quantity) as a function of photon wavelength. There will be a broad thermal peak somewhere ...


19

The theorem is called the noiseless coding theorem, and it is often proven in clunky ways in information theory books. The point of the theorem is to calculate the minimum number of bits per variable you need to encode the values of N identical random variables chosen from $1...K$ whose probabilities of having a value $i$ between $1$ and $K$ is $p_i$. The ...


18

UPDATE: Below I am answering yes to the first question in the post (are the two kinds of entropy the same up to a constant). This has led to some confusion as both Matt and John gave answers saying "the answer is no", however I believe they are referring to the title "Does entropy measure extractable work?". Although the author uses the two interchangeably, ...


17

Statistical Mechanics is the theory of the physical behaviour of macroscopic systems starting from a knowledge of the microscopic forces between the constituent particles. The theory of the relations between various macroscopic observables such as temperature, volume, pressure, magnetization and polarization of a system is called thermodynamics. first ...


17

From a fundamental (i.e., statistical mechanics) point of view, the physically relevant parameter is coldness = inverse temperature $\beta=1/k_BT$. This changes continuously. If it passes from a positive value through zero to a negative value, the temperature changes from very large positive to infinite (with indefinite sign) to very large negative. ...


17

The realization of non-Abelian statistics in condensed matter systems was first proposed in the following two papers. G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991) X.-G. Wen, Phys. Rev. Lett. 66, 802 (1991) Zhenghan Wang and I wrote a review article to explain FQH state (include non-Abelian FQH state) to mathematicians, which include the explanations ...


17

This is a very interesting question which is usually overlooked. First of all, saying that "large scale physics is decoupled from the small-scale" is somewhat misleading, as indeed the renormalization group (RG) [in the Wilsonian sense, the only one I will use] tells us how to relate the small scale to the large scale ! But usually what people mean by that ...


17

If the particles are not point-like, they will take up some volume. As the gas is compressed, the collision frequency will rise more quickly, which will make the pressure-volume curve change. The corrections in the Van der Waals model of a real gas accounts for the volume of the particles. Also if they have internal structure, that structure can have ...


14

If you were to surround the atmosphere by an adiabatic envelope and allow it to come to equilibrium, it probably would settle into such a state. However, the atmosphere is not a static place. It is actively mixed due to heating of the ground by the sun, and by cooling of the upper atmosphere by radiation into space. This makes the surface air less dense than ...


13

I don't really see the answer in the other answer so let me do the calculation here. Your general Boltzmann Ansatz says that the probability of a state $n$ depends on its energy as $$ p_n = C \exp(-\beta E_n) $$ where $\beta = 1/kT$. Fermions are identical particles that, for each "box" or one-particle state they can occupy (given e.g. by $nlms$ in the case ...


13

A convenient operational definition of temperature is that it is a measure of the average translational kinetic energy associated with the disordered microscopic motion of atoms and molecules. The underlying framework of all matter is quantum mechanical. This means that the Heisenberg Uncertainty principle holds. Even for a single particle the HUP ...


13

DavePhD's answer explains the specifics. The separation decreases the enthalpy of the oil-water mixture. But there's one more step: When the enthalphy of the dressing decreases by $\Delta H$, it causes the entropy of the dressing and its surrounding environment to increase by $\Delta H / T$. The reason is: The decrease in enthalpy releases heat, which ...


12

Another use of the SLE approach seems to be (I haven't read the papers below much beyond their abstracts) as a tool to probe for the presence of conformal invariance in various systems, when a direct (numerical or experimental) verification is difficult. In this approach, (i) one extracts suitable non self-crossing paths, (ii) one determines (empirically) ...


12

The Ising model is a model, originally developed to describe ferromagnetism, but subsequently extended to more problems. Basically, it is an interaction model for spins. Imagine you have a system which is a collection of $N$ spins. Each spin $S_i$ has two possible states $+1$ or $-1$. Here you can imagine already a possible extension to more states. You can ...


12

I guess so - I mean, as far as I know, there's no law of physics that strictly prohibits those "exotic" states from being realized. As long as the state exists and can be reached by some path from the "center" of the state space where the likely states are, there should be a nonzero (not even infinitesimal, really) probability of accessing it. But for a ...


11

Hannesh, you are correct that the second law of thermodynamics only describes what is most likely to happen in macroscopic systems, rather than what has to happen. It is true that a system may spontaneously decrease its entropy over some time period, with a small but non-zero probability. However, the probability of this happening over and over again tends ...


11

There's no temperature. If we use the following definition "temperature is the average kinetic energy of the particles". Then no particles - no temperature. As the first sight this answer doesn't seem to be good enough, but if you want to calculate "average spin" or "average charge" those parameters will have no sense if there's no particles to calculate ...


11

SLEs can be used in order to define a certain kind of QFT. You can check M. Douglas' talk, from page 28 forward: Foundations of Quantum Field Theory (PDF). There's also another great article, Conformal invariance and 2D statistical physics. You may also like SLE and the free field: Partition functions and couplings. Finally, there's an approach to try and ...


11

First of all we must distinguish between two things that are called entropies. There's a microscopic entropy, also called Shannon Entropy, that is a functional over the possible probability distributions you can assign for a given system: $\displaystyle H[p] = -\sum_{x \in \mathcal{X}}\; p(x) \log(p(x))$ where $\mathcal{X}$ is the set where your variable ...


11

Here's a self-contained (hopefully clear) derivation. Step 1. Setup and definition of differential scattering cross-section Let $\mathcal L$ denote the incident luminosity (number of incident particles per unit area, per unit time) of a beam to be scattered. We assume that we have a spherical detector at infinity with which to measure scattered particles, ...



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