# Tag Info

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I am going to focus on the precise meaning of the density matrix and the probability distribution $w$ in Landau's notation, which could be called the quantum statistical distribution. Almost all texts and wikipedia mess this up. If the usual axioms of quantum mechanics are accepted, every system is in a pure state, given by a wavefunction. Sometimes this ...

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The von Neumann entropy, written in terms of the quantum mechanical density operator, is a constant of the motion if you keep track of everything (including entanglement with the environment) and don't have any collapse events (which, depending on your favorite interpretation of quantum mechanics, might not exist anyway). The thing is that this fact already ...

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What books usually call "thermo-dynamics" is really "thermo-statics", the time variable is missing. There are all kinds of steady states and ever since Kirchhoff (~160 years) people have been trying to find some extremum principle to describe these. Kirchhoff showed that stationary electric current is distributed so that the dissipation is minimized and ...

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When normalized, $A$ is just equal to $1,$ so that $f(E)$ varies between $0<f(E)<1.$ Addendum for the edited question: The prefactor $\frac{2}{(2\pi\hbar)^3}$ crops up in the volume integration of density of states performed in k-space for the computation of number of states $N$ (i.e. all available energy states up to a certain maximum (fermi level) ...

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I think you should look into Molecular Dynamics software. GROMACS is open source and capable of doing a simulation like the one in the video. Disadvantages: Installation on Windows is tricky, lots of learning to get started (no GUI, long text files…) you need an extra programm to view your output as video… Advantages: Very flexible: Differert Algorithms ...

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The Second Law of Thermodynamics is an approximation, it has statistical or probabilistic validity. Statistical Mechanics corrects the plain flat out version of it that says entropy never decreases to the following. The overwhelming majority of the time, a sufficiently large system which is not a closed system (in the sense of mechanics: note that in ...

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Landau--Lifschitz is not reliable. I never recommend it to someone who does not already know the basics. It can often be inspiring....to an advanced physicist. This "proof" if a famous fallacy. It relies on his argument that the distribution function of the combination of two independent subsystems is the product of their individual distribution ...

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The first thing to point out is that there are two equivalent ways to describe a quantum statistical distribution: the density matrix, and a probability distribution on the results of measurements of a complete commuting set of variables. Any set will do. (Remarkably, it is also equivalent to give a probability distribution on the classical phase space of ...

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Steady state, as can be seen from the meaning of the word "steady" is "steady" in some way. Usually, there are multiple properties of the system that become steady or constant in time. For a thermodynamic system, these properties can be energy, entropy, temperature, pressure, volume. Then, it seems impossible to talk about a non-equilibirum system that is ...

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Regarding to your logic, the total amount of electrons at states with E2 and E3 altogether could be bigger than at ground state (with E1). It is inversion, isn't so? Thus the thermal invertion could be realized in three-level scheme but it is impossible at two-level system.

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I'll take a stab at this, although I am not an expert in the laser fields. Negative temperatures likely do not work: One concept that may seem closely related to the possible "thermal excitation of a laser" is that of "negative temperature". In a state of negative temperature higher energy levels have higher occupation probabilities than lower ...

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Hamurabi, I assume you are asking for the order of the wave equation. Try thinking about intramolecular vs. intermolecular forces. A wave equation is necessary to describe the energy of any quantum occurrence, so far as I recall. It may take an approximation for any non-ideal system, but exact or not the wave equation describes the moment in time when the ...

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The population over the rotational levels is a function of temperature in accordance to Boltzmann's law. For low temperatures, only the lowest rotational levels are populated while for higher temperatures more and more rotational levels get populated and the rotational energy increases accordingly. Note that in the high temperature limit you can replace the ...

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The pair correlation function $g(r)$ is defined via the density-density correlation function $$S_{nn}(r) =\langle n(0) n(r)\rangle .$$ Typically, we define $g(r)=S_{nn}(r)/\langle n\rangle^2$. The density $\delta n = n-n_{cr}$ is also the order parameter for the critical point on the liquid-gas phase transition. This means that the correlation function of ...

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The minimal counterexample seems to me to be the following: Take two materials, placed next to each other: ____________________ | | | | Material|Material | | 1 | 2 | ____________________ E1 _ _ _ E0 _ _ They have energy levels as indicated above- both have states at E0 and E1, but one has two excited states. ...

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To find the global minima of a function in a configuration space using Monte Carlo methods there are two main approaches simulated annealing and parallel tempering. Simulated annealing Simulated annealing is single Markov chain starting at high temperature for global exploration. The system is then evolved via Monte Carlo update whose criteria for ...

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A infinite barrier of potential reflects the fact the particle cannot enter a certain region of space. Solving the Langevin equation with such a barrier means that you have to find a way to state that the particle cannot enter the domain, but you also have to describe what happens at the boundary, because several scenarios are possible : the particle stops ...

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Here may not be a complete answer, but it may give you (and me) some hints and perhaps some alternative solution. Euler's scheme should not work, even for the deterministic equation where noise is set to zero. This is because in Euler's scheme, one always requires $\Delta t$ small so that the $\Delta x$ is small. When $\Delta x$ is large, one runs into the ...

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Therefore, when we say, for example, that the energy of the ideal gas at temperature T is E=32NkBT, we should really be saying "the energy of the ideal gas immersed in a heat bath at temperature T"? Is this reasoning valid? This is true. What is also true is that you can also say that the temperature of the (completely isolated) gas of energy E is T=2/3 ...

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From Reif's derivation of the canonical ensemble, the idea in describing the smaller system S is to use the fact that the total system including both S and R is isolated and in thermal equilibrium. That is, the total energy, using the notation you have written, would be $E_T=E_1+E_2$ and it must be constant. Then we use the assumption that, for this total ...

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The acceleration of gravity is the weight of the object on the surface. If the structure, either of the object or the surface, is deformed by the weight, then part of the gravitational potential energy of the object will turn into kinetic energy of surface moleucles and therefore heat, and this will be shared between the object and the surface. In this ...

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With respect to current theory my answer may prove quite unpopular (down votes expected), so please take this answer with a grain of salt if you are still in school. Your approach in this 'thought experiment' that you propose is very much like some of my own research conclusions. In direct answer to your question, yes, General Relativity should require an ...

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I think there is some confusion of terminology here.. A system is not at thermal equilibrium if you lock it into the most probable state. A single state is not an equilibrium, it is the complete evolution of the system which has to be observed (or known) to say if it's in equilibrium. If we take your A,B,C example, the system will be in thermal equilibrium ...

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It depends if we look at particle as classical ones or quantum ones. In the first case, particles are usually following a Boltzmann statistics. However, things become more interesting when entering the quantum world. Here, the spin of the particles become crucial. We have that particles with integer spin follow a Bose-Einstein statistics. Whereas particles ...

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The initial state does not need to be one of eigenstates of the hamiltonian, it could be superposition. Therefore time evolution will change it. I don't think your first assumption is correct.

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A body at rest in a gravitational field ((as standing on the surface of the earth) is the same as a body being accelerated. A free falling object is equivalent to not accelerating (The equivalency principal). The Accelerated one at rest would heat up. OK so maybe I didn't articulate that so well please let me try again. The equivalence principle states that ...

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If the object is at rest it would imply that gravity transfer heat without an increase in potential energy, and there are no other forces that produce work. This would violate the conservation of energy. Regarding your comment "any contact forces imposed on an object will increase that object's heat energy": this is incorrect, friction only results in heat ...

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It is just what is needed to describe a pure chemical substance. For other systems you may have fewer or more than 3 degrees of freedom - yes, many dof may occur in practice, think of a complex mixture of chemicals such as in crude oil. The general case is usually discussed in the context of Gibbs' phase rule.

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The assumption of a full eigensystem is usually made for convenience. But it is not always satisfied. If it is not satisfies one gets additional logarithmic contributions to the scaling laws. This is discussed, e.g., in the paper by Wegner and Riedel.

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As Christoph said, entropy is a measure of microscopic freedom. This is, I think, better than "measure of energy dispersal" (a now common description), for two reasons (and each one is enough for me): 1/ freedom can be measured in bits (or nats, digits, ...). 1 bit of freedom is the freedom to choose one of 2 options. entropy measures the freedom, the number ...

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It's not the rapid decrease in pressure, but the associated turbulence. There's a pinch point in the valve that the beer squeezes past, and in the process undergoes lots of turbulence. Like shaking a soda can, this causes the $CO_2$ to come out of solution as froth. Adding the long hose adds drag to the flowing beer, slowing its motion through that pinch ...

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Disclaimer: just so you know the following analysis assumes the compartments are well-mixed (diffusion is fast) which is likely a poor assumption, but it makes the analysis doable within engineering accuracy. So first off lets confirm that indeed the steady-state concentration of $B$ is what you say it is. Since there is no reaction between $A$ and $B$, we ...

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At the risk of being too obvious, let me first point out: any state that changes with time is not an eigenstate of the Hamiltonian. So if you are describing a system that is at equilibrium for all time, then you may indeed assume that system A is in an eigenstate of the many-body Hamiltonian, but for any other situation (including anything in the real ...

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Here's how I would come to some intuition for it. I would think about the rate of "probability flow" into a region by integrating the equation over a region in space. For now, let's suppose that no diffusion occurs at all, since that is more complicated (although directly doable and understandable). Then $$\int_a^b dx\frac{\partial p(x,t|x_0)}{\partial ... 2 I think you make it sound much more mysterious than it is. The relativistic distribution function is$$ f_p = \frac{1}{(2\pi)^3}\exp(-(\mu-u\cdot p)/T)\, $$where u_\alpha is the 4-velocity of the fluid, p_\alpha is the 4-momentum of the particle, T is temperature, and \mu is the chemical potential. This is sometimes called the Juttner ... 0 The ideal gas law is a cpmbonation of many other laws about gases. Some assune the pressure to be constant, others assume the quantity stays constant and others. Now those laws have been set up mostly after experiment and it people working on it noticed that the pressure P according to the small laws seemed to be proportional to the quantity n (in ... 0 In classical thermodynamics, temperature T is defined through ideal gas equation$$pV = nRT$$from which we conclude that$$K.E. = \frac{3}{2}k_BT$$is true for any ideal monatomic gas which cannot exist in real life anyways. Statistical mechanics provides postulates that is broader in context. It redefines the temperature through the second law$$dE = ...

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The electric fan increases the velocity and hence the kinetic energy of the molecules in the air. this would mean that the temperature has increased. I think that there is a bit of a problem here. Kinetic energy is not quite the same as thermal energy and temperature. Thermal energy and temperature is a measure of random, thermal motion of atoms or ...

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What you have written are all correct. But we should note that $P(E)$ alone is not the partition function. The canonical partition function is $$Z \propto \Omega_\mathrm{tot}(E_\mathrm{tot}),\qquad (1)$$ which counts the microstates of the univserse, i.e., system and bath. But $P(E)$ only counts the microstates of the bath! Let us see how to use the ...

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A typical renormalization group flow can be thought of as a smooth vector field $\vec V(\mu)$ defined on parameter space. Starting with parameters $\vec\mu(\ell)$ at scale $\ell$, you obtain parameters at scale $\ell'$ by solving the differential equation $\frac{d\vec\mu}{d\ell}=\vec V(\vec\mu(\ell))$. The function $R$ referred to above can be thought of as ...

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I'm not sure this is quite what you're looking for, as classical statistical mechanics in equilibrium is, by definition, not a properly dynamical system, but min/maxing the function $$S[p_i] = -k_{B}\sum_{i}{\ln{p(x)}} + Z\sum_i{\big(p(x)-1\big)}+\beta\sum_i{p_iE_i-\langle E\rangle}+\mu\sum_i{N_ip_i-\langle N\rangle}+...$$ not only recovers the Boltzmann ...

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The simplest way to resolve this paradox is to require $$\rho_B(t=0)\simeq \tilde{\rho}_B=\frac{1}{Z}e^{-\beta H_B}.$$ That is to say, you do not need any time evolution to reach thermal equilibrium, and this is the statement of ETH. This is very different from a classical chaotic system, where you need some time to explore the phase space and reach ...

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