Or to make it simpler just change t into -t in a parabolic type equation, e.g. heat conduction equation? Will it become an ill-posed problem? Will it have a physical meaning? I came up with 1 example when it has for the Stefan-type problem with phase changes.
There is very well meaning in inspecting the behavior under time inversion. While the fundamental microscopic dynamic equations such as F=ma are invariant under time inversion, those equation which reflect the impact of the second law of thermodynamics are not. Examples are Ohm's law j= sigma E which is NOT invariant under time inversion (because it implies the dissipation of electric energy into heat) and likewise heat transport is NOT invariant under time inversion. It reflects Clausius' version of the second law and the arrow of time determined by the second law of thermodynamics.
There is very well meaning in inspecting the behavior under time inversion. While the fundamental microscopic dynamic equations such as F=ma are invariant under time inversion, those equation which reflect the impact of the second law of thermodynamics are not. Examples are Ohm's law j= sigma E which is NOT invariant under time inversion (because it implies the dissipation of electric energy into heat) and likewise heat transport is NOT invariant under time inversion. It reflects Clausius' version of the second law and the arrow of time determined by the second law of thermodynamics.
Dear Dr. Chien-Lu Ping, you probably meant "in the 4th dimension"?
I would also add to the discussion above that we do not need to refer to religion to consider the possibility that time indeed is an illusion. Many physicists would argue that time is not fundamental but an effective parameter somewhat similar to the way in which temperature is not fundamental. A hint in this direction may be the fact that the concept of a time operator is very problematic in quantum mechanics.
Natalia, do you mean hysteresis effects? Formally you cannot reverse time in classical Stefan problem. May be better to formulate question in a way which is more familiar with the definite problems?
Natalia, I am sure the above tangential responses are not what you looked for. The simplest case studies of parabolic equations would be diffusion or the heat equation. Indeed, if you naively reverse time you run into a problem in the sense that the spatial dimensions become imaginary, which is not physical. The best way to understand it is by going back to basics and see how the equations are derived. Most intuitive is the diffusion equation. It is derived by using both conservation of mass - the time evolution of the density f within a given volume and the flux of mass out of this volume are constant - and a constitutive relation - the flux of mass, J, is proportional to the gradient of the density, up to a scalar function. It is only the former that involves time: df/dt + div(J) = 0. Reversing time in this equation has a straightforward interpretation: mass is changing flow direction into the volume instead of outward or vice versa.
The second equation only establishes that the flux is in the direction of decreasing density and it is independent of the direction of time, namely, it is true whether time runs forward or backward.
Bottom line, it is the combination of these two equations that gives you the parabolic form, but only one of them really involves time. The same physical interpretation applies to all parabolic forms that I know of.
I hope this helps. If you wish to discuss further your specific problem I would be glad to assist.
Dear Raphael and Pavel, this problem emerges in mathematical modeling amorphous to crystalline ice phase transition, which goes to higher temperatures with the release of latent heat. Usually it is the opposite. (Also, this unusual phase change happens in the range of temperatures and is irreversible.)
@Pavel and Raphael:
For that reason: - Latent heat, we needed to" hide" the "-" in the main eq.
It seems that the time variable is not a good way to do it. We just didn't want to use an explicit finite difference scheme with the " - latent heat" as a source function in the main eq.
We can reverse time for the Stefan problem. Example pls see in my paper in RG with Khodyko and Saverchenko. Also, we have to keep in mind that instead of evaporation we have to consider the condensation.
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