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I'm not a 100% this will address the question, so this might be more of a long comment than an answer. My main point is, perhaps, that given a Hamiltonian, you'll still want to specify the dynamics to do simulations. Even the Ising model can be simulated in several different ways, all satisfying detailed balance, where the relaxation towards equilibrium is ...


3

You are correct in your interpretation that Weisner's method is geometric in nature: it is a method for finding generating functions for special functions using representation theory of Lie groups and Lie algebras. And as you know, Lie groups play an enormous role in modern geometry, on several different levels. Lie groups are smooth differentiable ...


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You might be best off creating a bunch of trial systems and evolving each one independently using the (much easier) Hamilton's equations, which are good old ordinary differential equations. After all this is how the Liouville equation was constructed in the first place by people like Gibbs---in the limit of an infinite number of trial systems, you get the ...


4

You basically have the right idea: the existence of a covariantly constant vector field is a big restriction on the metric. You already discovered that $A^a$ has constant norm. The next thing we find is that $A^a$ is a Killing vector, because obviously $$\nabla_{(a}A_{b)}=0.$$ Furthermore, $A^a$ is geodesic, $A^a\nabla_a A^b=0$, and hypersurface ...


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You should use complex numbers. Assume the solution is of the form $y_i(t)=\tilde Y_i\mathrm e^{\mathrm i\omega t}$ where $\tilde Y_i$ is complex. Write also $\tilde F_i=F_i\mathrm e^{\mathrm i\phi_i}$ such that $F_i(t)=\mathrm{Re}(\tilde F_i\mathrm e^{\mathrm i\omega t})$. The system of equations becomes "simply" (it's "only" of the fourth order) $$ ...


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Suppose you have 5 cells with 1 ghost cell (related to your particular boundary conditions, whatever they may be) on either side. In this case, you can write your steady-state heat equation as \begin{align} T_0-2T_1+T_2 &= dx^2\,f(T_1)\\ T_1-2T_2+T_3 &= dx^2\,f(T_2)\\ T_2-2T_3+T_4 &= dx^2\,f(T_3)\tag{1}\\ T_3-2T_4+T_5 &= dx^2\,f(T_4)\\ ...



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