New answers tagged

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You said it yourself the molecules have direction rather than randomly moving about. Picture a wide spot in a river, slow water, maybe eddy currents, random water flow. River narrows, water is directed through the slot. I agree with you, speed before pressure differential. What do you think about this, molecules entering Venturi creating vacuum which sucks ...


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I have went through a Thermodynamics Course, one of the best book which had used to clear my problem is: Thermal Physics by " Michael Sprackling" it describe the entropy in terms of both "Microstate" and "Macrostate". I have lot of doubts and which was very helpful for me


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I'll answer my own question. The hint I've added in the end brings on the right path: in fact, even in the situation in which all the system's particles are stored in the reservoir, the integral $\mathscr Q = 1/ N! h^{3N} \int e^{\beta H_R} dp_R dq_R$ is performed just on the volume $V_R$, so the proper way to write the equation above is $\mathscr Q = 1/ N! ...


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Entropy Demystified (The Second Law Reduced to Plain Common Sense) by Arieh Ben-Naim. Authored discussed not only the thermodynamics origin of entropy but also the same notion in the context of information theory developed by Claude Shannon.


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The evolution of the system is govern by the Hamiltonian. Let the system is evolved from the initial state $\rho_{i}=\frac{1}{Z}\exp^{-\alpha H(t)}$. Here $Z=\mbox{Tr}\exp^{-\alpha H(t)}$ and $\alpha$ is the temperature of the initial state. Suppose the Hamiltonian $H(t)$ is varying in time. If $\rho_{f}=\frac{\exp^{-\beta H(t\rightarrow\infty)}}{\mbox{Tr} ...


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Consider the probability of state $n$ divided by the probability of state 0: $$\frac{\text{prob of }\left \lvert E_n \right \rangle}{\text{prob of }\left \lvert E_0 \right \rangle} = \frac{\exp \left( -E_n / k T \right)}{\exp \left( - E_0 / k T \right)} = \exp \left( - (E_n - E_0) / kT \right) \, .$$ If $E_n > E_0$, then as $T \rightarrow 0$ the ...


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The derivation by Pols is correct. Ryden makes the strange decision to plug the relativistic rest energy $\varepsilon = \rho c^2$ into the classical ideal gas law. Surely it makes more sense to define a classical kinetic energy $$ u = \frac{1}{2}\rho\langle v^2\rangle $$ so that $$ P = \frac{2kTu}{\mu\langle v^2\rangle} = \frac{2}{3}u. $$


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I figure that given the rather extensive question that it does deserve some answer. I don't know if there is stack exchange on the history of science or physics, but that might in fact be a more appropriate place for this. The grandfather of quantum mechanics was Max Planck. His assumption that distributions of energy occurred in discrete units is what lead ...


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It has to be " Statistical Mechanics and Thermodynamics " by Claude Garrod". You can use the text by Macquarie as a supplement. For renormalization group and advanced concepts, use " Statistical Physics of Fields" by Kardar.


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A system is in a heat bath of temperature T so we work with the canonical ensemble. We consider $N$ degrees of freedom $x_1, x_2, ..., x_N$ and $x$ is the vector $(x_1~ x_2 ~ ... ~ x_N)^T$. The potential energy is quadratic so it can be expressed as a function of its second derivatives: $ U=\sum_{i,j} x_i ~H_{i,j} ~x_j = x^T H x ~~ $ with $H_{i,j}=\frac{\...


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I have been wondering the same question recently. There is a way to define the isobaric partition function that seems to work pretty well with large systems, but I'm not convinced that it's appropriate with small ones. The idea is not to define a measure of the volume; instead define volume indirectly via the canonical ensemble, as follows. In the canonical ...


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The question may be about the covariance or otherwise of temperature, in which case have a look here. As well, have a look at the paper "Temperature in special relativity" by J. Lindhard, Physica Volume 38, Issue 4, 5 June 1968, Pages 635-640.


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Evaporation depends on partial vapor pressure. So the control parameters are, - temperature - partial pressure - vapor pressure at the temperature For the hot tub that you are studying, humidity will slow down evaporation and dry will help. Cold is not good for evaporation. In reality, you may not be able to make all happens at the same time.


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In the kinetic theory of gases, you only really define the temperature for molecules that are in constant, random, and rapid motion. So if you have a container with a gas at temperature $T$ you don't change the internal energy of the gas by uniformly moving the container. Uniformly moving the container gives all the molecules a non-zero average motion, but ...


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Statistical mechanics offers us different tools for different types of systems. The microcanonical ensemble describes a system $S$ with fixed energy $U$, particle number $N$, and volume $V$. The canonical ensemble consists of a reservoir $R$ with fixed volume $V_R$, particle number $N_R$, and temperature $T_R$ and a system $S$ with fixed volume $V_S$ and ...


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In a sense, you can consider chromatography to be a kind of 'filter', and it can certainly separate out coffee components, including water. There's no lack of other ways, though, to extract water. Freeze drying would carry the water vapor away, and you can sell the residue as instant coffee...


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The density of states is defined that way, such that in a continuous distribution (which is valid here because the energy is large), the number of states between $E_1$ and $E_2$ is $$N = \int_{E_1}^{E_2}dE \ \Omega(E)$$ where $\Omega(E)$ is the number of microstates for energy $E$. In the case of the infinitesimal range $\delta E$, the function varies ...


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I will address the stat mech part. Over-counting refers only to indistinguishable particles, i.e. those with all the same properties. It also only comes into play when you are performing calculations by "labeling" or "tracking" indistinguishable particles. This is not the same as using the number of particles in each energy state. In other words, if you ...


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Many-body systems and other complex phenomena exhibit what is known as emergence. If I may reformulate your question, then you are essentially asking why this is the case. In some generic situations, the answer can be simple. A very old argument, which was probably already known at the advent of statistical physics is the following (you might find it also ...


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I think you can avoid all these troubles if you define the temperature as proportional to the variance of velocity, i.e. $$E[(v-\overline{v})\cdot(v-\overline{v})]=E(v\cdot v)-\overline{v}\cdot\overline{v}$$ Here $E$ means expected value, $v$ ranges over the velocities of the individual particles, and $\overline{v}=E(v)$. Clearly this is frame-...


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All kinds of weird things happen if you try to define temperature in a moving object. The paradox to me (not a generalized accepted answer) resolves by realizing that temperature should only be defined as measured when the object is stationary. Not only is not a scalar but it is not even well defined for areference frame in relative motion. Is temperature ...


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Coffee is a homogeneous solution hence it can not be separated by usual methods. I have few suggestions may not be very accurate but good for brainstorming. Distillation : Although you have mentioned not to mention it but I would like to add that coffee has several aromatic organic compounds that make the smell of coffee hence you can not get rid of ...


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My guess is that the volume density of states is 1 per unit volume, and the (energy?) probability density is probably N!/(n1!n2!), and then the author took an integral over those times dE, said "these are constant" and threw in the wiggly dE (to replace the integral being multiplied by 1*N!/(n1!n2!)). (This is supposed to be a comment)


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Temperature is related to kinetic energy in the rest frame of the fluid/gas. In non-relatvistic kinetic theory the distribution function is $$ f(p) \sim \exp\left(-\frac{(\vec{p}-m\vec{u})^2}{2mT}\right) $$ where $\vec{u}$ is the local fluid velocity. The velocity can be found by demanding that the mean momentum in the local rest frame is zero. Then $\vec{u}...


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The generalized equipartition theorem states that if $x_i$ is a canonical variable (position or momentum variable), then $$\left\langle x_i \frac{\partial \mathcal{H}}{\partial x_j}\right\rangle = \delta_{ij}\ k T$$ where the average $\langle \cdot \rangle$ is taken over an equilibrium probability density $\rho(p,q)$: $$\langle f(p,q) \rangle = \int dp dq ...


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Equipartition of energy only holds and for a system in thermodynamic equilibrium. The generalized equipartition theorem states that if $x_i$ is a canonical variable (position or momentum variable), then $$\left\langle x_i \frac{\partial \mathcal{H}}{\partial x_j}\right\rangle = \delta_{ij}\ k T$$ where the average $\langle \cdot \rangle$ is taken over an ...


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[June 19,2016: thoroughly revised, giving a more detailed, comparative presentation and better references] General case. In relativistic thermodynamics, inverse temperature $\beta^\mu$ is a vector field, namely the multipliers of the 4-momentum density in the exponent of the density operator specifying the system in terms of statistical mechanics, using the ...


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If you want to filter out the grains then certainly you could using normal filter papers in a filter funnel and repeat until the solution is clear of bits. You could also use a sintered glass filter. However, there will still be compounds from the coffee dissolved in the water and so a molecular sieve could be used and/or a chromatography column to separate ...


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I'd think of effusion effect, centrifugal techniques or other methods used for uranium enrichment. You could possibly select particles with the right mass (mass of the water) out of this mixture in a lot of iterations. It would be long and very expensive but probably possible.


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A Boltzmann brain is not that different from what might be called Boltzmann cheese. Given enough time a set of atoms or particles might arrange themselves by statistical fluctuations into big wheel of cheese. If that happens there is no reason to think the cheese would then rapidly be demolished unless it formed in a star, or falling into a black hole or in ...


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I just want to add a second option that was proposed by some author whose name I cannot remember (it might be wikipedia, I did not check). Perhaps the laws of physics are such that it is very difficult to form a spontaneous and isolated boltzmann brain. According to this idea the only way to form a brain is to first spontaneously form a universe, like ours, ...


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After quite some thought, I have come to the conclusion that my question is more a mathematical question than a physical one. What I am looking for is actually a formula for a change of variables for a probability density function. For example, if we have the random variable $X$ with probability density $\rho_X(x)$ and another random variable $Y=f(X)$, ...


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I would not say it decays instantaneously, but it will decays quickly, at least the conscious part. There is no body that supports it, so it will lack the oxygen to think in a brief moment. The rest will be bombarded by radiation and other particles until it disintegrates again, the timescale for this to occur will be a function of the temperature.


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The Boltzmann brain paradox arises due to smaller fluctuations being more probable than larger ones. So, if you contemplate how our universe started out with low entropy initial conditions then it's difficult to explain this in terms of a generic high entropy state. Fluctuation yielding the early universe that in turn would have given rise to you, would be ...


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Writing the partition function in terms of the density of states boils down to simply ordering our summations to count up all the states with the same energy first, and then summing over the different possible energies. This procedure can be done regardless of the coordinates being used to describe the states; there is no need to introduce an effective ...


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This combination represents the following. $f$ in your reference is the number of particles per cubic space volume per cubic velocity volume. When you multiply it by some volume in velocity space, you obtain the regular density — number of particles per volume. But, as $f$ itself depends on velocity, to obtain the total density of all particles with all ...


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The problem when you include gravity or other long range forces, is that thermodynamics becomes non extensive. For instance, the energy of the union of two systems is not the sum of the energies of the individual systems. To handle those cases, generalized entropies have been proposed. By generalized it means that these formalisms allow for long range ...


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I never saw a proof and don't know where to find one. Infinitely stable substances only exist according to a simplified quantum mechanical theory where gravity and nuclear chemistry don't exist. Whether or not there has been a published proof that proves it from that simplified quantum mechanical theory, I don't know. Maybe it has never been proven that the ...


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This is essentially a result of the equipartition theorem where each degree of freedom contributes $k_B T/2$ to the energy. Given that the specific heat in this context is just ${\partial E}/{\partial T}$ then each degree of freedom contributes $k_B/2$ to the specific heat. For the classical model of lattice vibrations in solids this leads to the Dulong-...


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I think it is wrong to define the temperature by the average energy of the molecule in all frames of reference. The reason for that is clear: take all of your particles and send them at $100 m/s$ to the north. This won't make the gas hotter, just like the fan does not cool/heat the air (another great mystery!). The organized movement does not participate in ...


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I am not sure if you're getting your concepts right !! In statistical mechanics, you don't study how the system evolves to a equilibrium state, you just take a system in equilibrium. In the Grand Canonical ensemble, the system is still in equilibrium although there is exchange of energy and particles. Lastly, the chemical potential and temperature are ...


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Leaving aside the issue of wavefunction collapse, physics is deterministic. So if you have some system like a gas and you know the exact positions and velocities of all the gas molecules you can predict the evolution of the system forwards and backwards in time. So you can start with a future state and work backwards to desciribe a past state. However ...


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Consider a cube with side length $a$. Then its perimeter is proportional to $a$, its surface area $A$ is proportional to $a^2$, and its volume $V$ is proportional to $a^3$. It's clear that any reasonable 3D shape satisfies the same kind of scaling, which implies $A \propto V^{2/3}$. However, it's possible for fractals to have perimeter/area/volume that ...


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I think there are a number of generalizations. The first is the partition function should be considered quantum mechanically. The reason is that spacetime can contribute to entropy, such as with Hawking radiation. To start the partition function is the trace $$ Z[\phi] = \sum_n\langle\phi_n|e^{-H(\phi)\beta}|\phi_n\rangle $$ Now, if one is interested in the ...


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Well I'll give it a try. I guess you have to be careful that after transforming you get two different quantized momenta. You're then summing/integrating over $q',q$. Leaving out the normalization constants: \begin{align} H[\tilde{\phi}] &= \int dx \left[k_1 \left( \partial_x \int dq \tilde{\phi}(q)e^{-iqx}\right)\left(\partial_x \int dq' \tilde{\phi}(q'...


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Let's think of the problem this way...it's a bit rough but should give you an idea: We know that the energy levels of a particle in a 1D box of length $L$ are given by the formula $$E_n = \frac{\pi^2 \hbar^2}{2 m L^2} n^2$$ where $n=1,2,3,\dots$. We also know that the average kinetic energy of a particle $\langle E\rangle$ is related to the temperature by ...


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The Fermi-Dirac distribution for a particle to be in the $n_i$ state with energy $E_i$ and $\mu$ the chemical potential equal to the Fermi energy as $T~\rightarrow~0$ is $$ \bar n_i = \frac{1}{e^{(E_i-\mu)\beta} + 1} $$ For the temperature $T~>>~0$ then $\beta~=~1/kT$ is small and for $E_i$ not large we have $(E_i-\mu)\beta << 0$. This means that ...


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Bayesian probability is based on the classical logic of plausible reasoning, as described by Jaynes, Probability Theory: The Logic of Science. One can and should try to find a Bayesian interpretation of the wave function. Here I can recommend Caticha's "entropic dynamics" as one possible approach. But I don't think "quantum logic" will be helpful here. ...


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A gas can be approximated as an ideal gas. You then assume that the particles don't feel each other and that they are infinitely small. The potential is zero. The particles can only have kinetic energy. If you would make a simulation of such a system the particles can literally move through each other. If you make this ideal gas approximation, it is ...


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You can model the gas a a collection of hard spheres of some radius $r$, and do the correction relative to the limit as $r\to\infty$ perturbatively. What you find is that for a fixed pressure and temperature amd number of molecules the first order correction to the law replaces $V$ with $V-\frac{4n\pi r^3}{3}$. This tells you that the volume to use is the ...



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