Search Results

The ergodic hypothesis is not, in general, true. It is true for some systems, for some initial conditions. The particular concern you raise is that there might be "sinks" in phase space - regions that …

Let time be periodic with period $1$. If you fix $\phi(0)$ (perhaps randomly) and your time evolution is deterministic, then $\phi(1)$ is already fully determined. A periodic time dimension implies $\ …

In order to ask if phase volume on the submanifold is conserved, we first need to define phase volume on the submanifold. It's not obvious how to do this - the symplectic form might vanish on the subm …

You ask when it is true that $$ \left\langle \frac{{p}_i^2}{2m_i} \right\rangle_{t} = \frac{1}{2}k_B \left(\frac{\partial S}{\partial E}\right)^{-1} \tag{1}$$ for a Hamiltonian system. If the Hamilton …

In order to use Wiener-Khintchine, we assumed that the process was stationary. This implies that $ \langle A_k(0)A_{k'}(0) \rangle = \langle A_k(t)A_{k'}(t) \rangle $ - our choice of initial time was …

Any $\rho$ which solves the equation on the whole torus must also be a solution locally on every subset. In particular, it must be solution on the (non-toroidal) open $L \times L $ square. Since solut …

The author's argument essentially reduces to a statement that $$\frac{d}{dt}\bigg|_{t' = 0} e^{-\gamma|t'|} = -\gamma $$ It should be clear to you that this is not true. The way he argues this is by …