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10

I once asked Putterman after a similar colloquium what he meant by this statement, and his answer was "long time tails". Long time tails are fractional powers that appear in the long time behavior of correlation functions, see, for example, here and here. These fractional powers are seen in molecular dynamics (they are more difficult to see experimentally), ...

16

They are derivable from classical mechanics using either the continuum or molecular points of view. Starting with a continuum view, one applies conservation of mass, momentum, and energy to a control volume and the result is the Navier Stokes equations. The Navier Stokes equations, in the usual form, apply to Newtonian fluids, that is fluids whose stress ...

20

None of the interesting equations in physics can be derived from simpler principles, because if they could they wouldn't give any new information. That is, those simpler principles would already fully describe the system. Any new equation, whether it's the Navier-Stokes equations, Einstein's equations, the Schrodinger equation, or whatever, must be ...

6

Nonlinear optical elements are called nonlinear precisely because of the behaviour you note: because the optical response of the material does not depend linearly on the driving fields. The response may then have a quadratic or higher dependence on the driver, which is usually written in the form $$\mathbf P =\varepsilon_0 \chi^{(1)} \mathbf E + ... 2 Non-Linearity means that the dispersion relation becomes non-linear. Linearity is an assumption which only holds true for low intensities. Almost every material has some non-linear effects if the light source is only powerful enough. The polarization vector for example becomes: P = P_0 + \varepsilon_0 \chi^{(1)} E + \varepsilon_0 \chi^{(2)} E^2 + ... 0 Here it explains what I think you are asking: We work with a formulation of Noether-symmetry analysis which uses the properties of infinitesimal point transformations in the space-time variables to establish the association between symmetries and conservation laws of a dynamical system. Here symmetries are expressed in the form of generators. We ... 1 I think I can remember the derivation for a conservative force field in classical mechanics, wich is a somewhat stronger assumption than pure time-translation invariance. Let \vec{F} be a conservative force field, that is$$ \nabla \times \vec{F} = 0 $$or alternatively$$ \phi := -\int_\gamma \vec{F} \cdot d\vec{a} $$does not depend on the path \gamma ... 0 I believe this is discussed in Nonlinear Dynamics and Chaos by Strogatz. I don't remember the details, but it's worth looking at his discussion of an energy function. 1 (After possibly introducing more variables) then OP is essentially considering an autonomous system of n coupled 1st order ODEs$$\tag{1} \frac{d\vec{z}(t)}{dt}~=\vec{f}(\vec{z}(t)), \qquad f: U \to \mathbb{R}^n , \qquad U\subseteq \mathbb{R}^n,  i.e. without explicit time dependence, so that the system (1) possesses time translation symmetry. OP is ...

0

Example of a hyperbolic system, the first order wave equation: ${\partial \underline{U} \over \partial t} + \underline{A} {\partial \underline{U} \over \partial x} = 0$ The term hyperbolic means that: The eigenvalues of the $m \times m$ Jacobian matrix ($\underline A$) are all real There is a corresponding set of $m$ linearly independent eigenvectors ...

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