What kind of PDE loses information in time?

[kof9595995]

Just learned that diffusion equation loses information as time goes on,i.e. given the initial condition at t=0, we can't uniquely determine the solution for t<0. And diffusion equation reminds me of Schrodinger equation, which looks very much like diffusion equation, except that the coefficient is imaginary. However, from my experience of solving it, Schrodinger's equation preserves information, initial condition at t=0 uniquely determine solutions at t<0. I'm wondering what's the criteria of determining whether a PDE loses information or not?

 

[jvicens]

Thank you for bringing up this interesting topic. I can provide some insight into your question about the criteria for determining whether a partial differential equation (PDE) loses information or not.

First, it is important to understand that PDEs are mathematical equations used to describe physical phenomena, such as diffusion or wave propagation. They involve multiple variables and their derivatives with respect to space and time. The solutions to PDEs are functions that describe the behavior of these physical systems.

Now, coming to your observation about the diffusion equation and Schrodinger equation, it is true that both these equations have similar mathematical forms. However, there are key differences between them that determine whether information is lost or preserved.

One of the main criteria for determining whether a PDE loses information is the presence of a dissipative term. A dissipative term is a mathematical term that represents the loss of energy or information in a physical system. In the case of the diffusion equation, the diffusion coefficient is a dissipative term, which means that as time goes on, the solution will spread out and lose its initial information. This is why we cannot uniquely determine the solution for t<0.

On the other hand, the Schrodinger equation does not have a dissipative term. The imaginary coefficient in the equation represents the quantum mechanical nature of the system, but it does not cause any loss of information. This is why the initial condition at t=0 can uniquely determine the solution at t<0.

In summary, the presence of a dissipative term is the main criteria for determining whether a PDE loses information. However, there may be other factors at play, such as the physical system being described and the boundary conditions imposed on the equation.

I hope this helps answer your question. If you have any further doubts, please feel free to ask. Let's continue to explore and learn together.