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If microstates are discrete then they can be counted. So in information theory, for example, we calculate the entropy of a message by counting the number of equivalent messages and comparing that with the total number of possible messages. However, if microstates are continuous then some other method of “counting” them is needed. This is typically done by ...


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The infinitesimal variation of a general action $S=\int_{t_i}^{t_f}\! dt~L $ is of the form $$ \delta S~=~\int_{t_i}^{t_f}\!dt \left(\frac{\delta S}{\delta q} \delta q +\frac{d}{dt}\left(\underbrace{p_j \delta q^j}_{=\Theta} \right)\right), \qquad p_j~=~\frac{\partial L}{\partial q^j}.$$ Here the boundary term $$ \Theta~=~p_j \delta q^j$$ is the pre-...


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The central idea is that the Dirac quantization can be implemented by replacing the Poisson brackets of functions $\{f,g\}$ with a commutator of operators $[\hat{f}, \hat{g}]$. So you need to know how to compute the Poisson bracket of two functions in the symplectic formulation of Hamiltonian mechanics. The central object in this formulation (see, for ...


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To expand on @probably_someone's comment, by setting $\dot{x}_1(0)=0=\dot{x}_2(0)$ you have already set the phase difference. Are a result, your solution is not the most general one. (One way to see this is that you only really have one free parameter left to "play" with, the phase difference $\phi_2 - \phi_1$. However the most general solution is ...


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It is a probability density, not a density of something like matter or energy. The probability density $f$ answers the question "how likely is it (what is the probability that) the microstate $\omega$ in one of the points in some set $A$": $$P(\omega\in A) = \int_A f(p,q)dpdq$$ The most other famous example of probability density in physics is the ...


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First off, the expression for the partition function in quantum mechanics can be expressed in terms of an integral. The partition function is $$Z = \text{Tr} \ e^{-\beta H}. $$ This trace can be evaluated in any basis. In the basis where the Hamiltonian is diagonal we write $Z = \sum_i e^{-\beta E_i}$, where $E_i$ are the eigenvalues of the energies and the ...


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The following assumption is used in some books of statistical mechanics at the very start and is not related to the canonical ensemble: "The phase can be split into small parts $C_i$ each with volume $h^3$ such that $H$ is constant on each $C_i$ and takes value $H_i$ on it". With this assumption, the continuous formula can be simplified : $$\frac{1}...


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TL;DR: The solutions to a first-order ODEs never intersect transversally as OP already noted. The absence of merging/splitting paths (i.e. intersecting tangentially) is guaranteed by the local uniqueness of first-order ODEs. A sufficient condition is that the evolutionary vector field $X_H$ should be Lipschitz continuous, cf. the Picard-Lindelöf theorem. ...


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You are right I think. The idea of attractors is similar to this. In case of attractors, you can start from any initial state and yet eventually evolve to a unique final state. From what I understand, it is only under some very special conditions when systems with vastly different initial condition tend to evolve into a unique final state. But that is not ...


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Your logic is correct. Often in classical mechanics, though, we restrict our attention to conservative forces, in which case the dynamics have time-reversal symmetry, that is, the system should have a unique phase-space path whether you go forwards or backwards. For merging paths, if the system is at a point past the merger and you reverse time, it's not ...


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The simple answer is, that that is what spacetime physically is, a topological space. It all started when Euler first counted the edges of a polyhedron and came up with his famous formula V - E + F = 2. When polyhedra or graphs drawn on surfaces like toroids, Klein bottles and Moebius strips came under study, they did not match his formula. The study of such ...


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This is a little more philosophy/mathematics than physics, but there is an alternative approach pioneered by Tim Maudlin that tries to replace point-set topology with a theory about straight lines. He takes the topological manifold approach to lack an intuitive basis, and so he proposes an alternative mathematical theory. It is called the theory of linear-...


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There have been a number of proposals motivated by quantum gravity to use non-Hausdorff manifolds to describe timeline splitting (as in the many worlds interpretation) and avoid time travel paradoxes. https://www.researchgate.net/publication/334223872_Interpreting_non-Hausdorff_generalized_manifolds_in_General_Relativity https://arxiv.org/abs/gr-qc/0505150


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Measurement instruments are not infinitely precise, however It is possibile to distinguish objects using them. This is possible when the precision of them permits it. The precision of an instrument around measured values is the physical corresponding of a neighborhood of a point. The fact that two measures can be distinguished by means of sufficiently ...


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But...you are in the wrong chapter! The equivalence inter-relations of the various ordering prescriptions are in Chapter 0.19 of that link! (Chapter 0.18 merely details the obvious Weyl transform (134)/(140) and its inverse, the Wigner transform (138) for all well-behaved phase-space functions, and all sensible Hilbert-space operators, of which the density ...


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TL;DR: The underlying basic identity behind the Bopp shift is a Taylor expansion, which amounts to a translation/shift, $$e^{\hat{A}\partial_x}f(x)~=~f(x+\hat{A})\tag{A}$$ Here we assume the operator $\hat{A}$ does not depend on $x$. Sketched proof: $$\begin{align} (f\star g)(x,p)~=~&\left. e^{\frac{i\hbar}{2}(\partial_p\partial_{x^{\prime}}-\partial_x\...


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There are some issues with the post on the website. I cannot understand how to pass form Navier-Stokes equations, which are PDEs, to some ODE system (Lorenz' equations?) which seem to be the subject of the discussion (also looking at the formulation of the question by Prof.Legolasov). Unfortunately I do not have access to the paper. From a pure mathematical ...


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Here is what is probably missing. That predictability limit required the existence of a feedback mechanism which operates during each time step in the simulation. The feedback cumulatively amplifies the sensitivity to initial conditions and magnifies the effects of rounding and discretization errors in the algorithms and thereby causes the system to go ...


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For the case of Wigner functions corresponding to pure states, I have been able to crudely determine the answer is "yes" based on some results in the very nice paper "Schwartz operators" (arXiv:1503.04086) by Keyl, Kiukas, & Werner (KKW). (Thanks very much to @cosmas-zachos for pointing me toward this!) I am pretty sure it must also ...


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Kronecker deltas are simply constants, being either $1$ or $0$ depending on the values of $i$ and $j$. An expression like $\{q_{i} + \delta q_{i}, p_{j} + \delta p_{j}\} = \delta_{ij} - \epsilon\{\delta_{ij}, g\}$ should really be thought of as $9$ different expressions for each possible value of $i$ and $j$, and in each of those expressions the Kronecker ...


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Start with the Poisson bracket $$ \frac{\partial Q_i}{\partial q_k}\frac{\partial P_j}{\partial p_k} -\frac{\partial Q_i}{\partial p_k}\frac{\partial P_j}{\partial q_k} $$ Then use the conditions you propose $$ \frac{\partial Q_i}{\partial q_k}\frac{\partial q_k}{\partial Q_j} +\frac{\partial Q_i}{\partial p_k}\frac{\partial p_k}{\partial Q_j} $$ undo the ...


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