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Jan
28
reviewed Approve What exactly “triggers” a matter/antimatter detonation?
Jan
22
comment Reference request: 2D conformal field theory and functions on the triangular lattice
You should google "discrete complex analysis". There are many papers introducing these ideas, for example those by Chelkak, Smirnov, etc.
Jan
22
comment Cluster Expansion
Done. I tried to make a readable answer out of the comments, but had only limited time, so there might be typos.
Jan
22
answered Cluster Expansion
Jan
22
comment Cluster Expansion
Yes you sum over all possible values of $n_\ell$ for each possible values of $\ell$ (so $\{n_\ell\}$ specifies the values of $n_1, n_2, n_3, \ldots$). Then, once these are fixed, you take the product of all the functions $f_\ell(n_\ell)$ (for these specific values of $n_1,n_2,n_3,\ldots$).
Jan
22
comment Cluster Expansion
Sure. Imagine that the only allowed cluster sizes are $\ell=1$ and $\ell=2$ (so that I can write things down explicitely). Let us use the notation $f_\ell(n_\ell) = \frac{1}{n_\ell!}\bigl( \frac{e^{\ell\beta\mu}b_\ell}{\lambda^{3\ell}\ell!} \bigr)^{n_\ell}$. Then your identity becomes $\sum_{\{n_1,n_2\} \in\{0,1,2,\ldots\}^2} \prod_{\ell=1}^2 f_\ell(n_\ell) = \sum_{n_1\geq 0} \sum_{n_2\geq 0} f_1(n_1) f_2(n_2)= \sum_{n_1\geq 0} f_1(n_1) \sum_{n_2\geq 0} f_2(n_2)= \prod_{\ell=1}^2 \sum_{n_\ell\geq 0} f_\ell(n_\ell)$
Jan
21
comment Cluster Expansion
This is just linearity of the sum. Maybe it will become obvious if you look at the following particular case: $\sum_{i,j} a_i b_j = \sum_i \sum_j a_i b_j = \Bigl(\sum_i a_i\Bigr)\Bigl(\sum_j b_j\Bigr)$, which follows by pulling out $a_i$ from the inner sum and then pulling out $\sum_j b_j$ from the outer sum. Now, in your formula $\{n_\ell\}$ denotes the number of clusters of each possible length (so $\sum_{\{n_\ell\}} = \sum_{n_1\geq 0}\sum_{n_2\geq 0}\cdots = \prod_{\ell\geq 1} \sum_{n_\ell\geq 0}$).
Jan
19
comment What are alternative ways to think about transfer matrix as used in Ising model?
@Nathaniel : yes, that's exactly the point. If you denote by $\varphi^{*,1}$ the left Perron-Frobenius eigenvector (associated, of course, to the same eigenvalue $\lambda_1$), then the only thing that changes is the expression for the invariant measure, which becomes $\mu( \sigma) = \varphi^{*,1}_\sigma \varphi^1_\sigma$. (Note also that you could define another Markov chain with transition probabilities $\pi^*(\sigma,\sigma') = \frac{\varphi^{*,1}_{\sigma'}}{\lambda_1\varphi^{*,1}_\sigma} T_{\sigma',\sigma}$; this corresponds to the time-reversed chain (which has the same invariant measure).
Jan
19
comment What are alternative ways to think about transfer matrix as used in Ising model?
@Nathaniel : thanks for your appreciation (and the unexpected bounty). Concerning the nonsymmetric case, I might either add a (short) second part to the answer, in which I point out the relevant changes, or simply point them out in the comments. What would be the best solution?
Jan
19
awarded  Mortarboard
Dec
29
comment annealed randomness vs quenched randomness
Quenched randomness means that the disorder is sampled first and then equilibrium statistical mechanics is applied to the system's other degrees of freedom with this particular frozen disorder. Annealed means that you average on the disorder at the same time as you take the thermal average over the other degrees of freedom.
Dec
7
comment What is the difference between toy models and normal models?
@Ooker : yes, it is a caricature, keeping enough relevant features to make it useful for the discussion of a particular aspect of the phenomenon, while keeping the model tractable enough to make a detailed analysis possible.
Dec
5
comment What is the difference between toy models and normal models?
@Ooker: the point of the Ising model is certainly not to provide quantitative information (even in a limit), but rather to provide a rather simple, yet very rich model in which a great variety of fundamental issues can be investigated. As a matter of fact, it does model adequatly a number of systems, but that's far from its main purpose. Universality also guarantees that it does provide a good description of the critical behavior of a wide class of (real) systems.
Dec
4
comment Extending the ergodic theorem to non-equilibrium systems
I am not at all an expert in these topics, but you might have a look at the book Mathematical Theory of Nonequilibrium Steady States, Lecture Notes in Mathematics, Volume 1833, 2004, and references therein. If you want something shorter, maybe look at this review by Ruelle.
Dec
4
comment Extending the ergodic theorem to non-equilibrium systems
@JánLalinský : I think that he's interested in "open" Hamiltonian systems, for which Liouville's theorem indeed does not hold.
Nov
28
comment Minimization of energy for non-equilibrium systems at steady state (NESS)?
No, it's not "a matter of definition", except if you wish for every physicist to use its own set of definitions for everything... There is a very clear definition of equilibrium systems, to which NESS do not belong: equilibrium systems are systems with time-invariant statistical properties and no macroscopic flow of matter or energy. The distinction is not gratuitous: the formalism of equilibrium statistical physics does not apply to NESS (and the same is true of equilibrium thermodynamics).
Nov
27
comment Minimization of energy for non-equilibrium systems at steady state (NESS)?
Yes, and @David, note that a lot of work has been done specifically for the type of stochastic dynamics that you seem to have in mind; see, e.g., this paper and papers citing the latter.
Nov
27
comment Minimization of energy for non-equilibrium systems at steady state (NESS)?
This is incorrect. There are of course nonequilibrium steady states (and these are studied so much that they have gained their own acronym: NESS). At equilibrium, there are no fluxes, no dissipation. In NESS, the system is submitted to a constant driving force (e.g., it is put in contact with two infinite reservoirs at different temperatures, or different chemical potentials, etc), which will induce fluxes in the system. What characterizes NESS among nonequilibrium states is that their statistical properties are time-invariant.
Nov
22
comment Can entropy be regarded as energy dispersal?
I agree that this is rubbish as a fundamental definition of entropy. Maybe it's useful to some people as a mental image of entropy, but it certainly has no general validity (except if you are ready to reinterpret in looser and looser ways the notions of energy and of dispersion, in order to suit each counterexample). Entropy is not a measure of disorder, nor is it a measure of energy dispersal. It has precise, general definitions both in Thermodynamics and in Statistical Physics. The latter may not be easy for students to grasp, but that does not mean that one should get rid of them...
Nov
19
revised What are alternative ways to think about transfer matrix as used in Ising model?
Added some (hopefully helpful) remarks. Simplified a bit the exposition (assuming the eigenvectors to be normalized).