# Time evolution of a microstate

In phase space each point represents a microstate of the system. And for a system in a certain macrostate, depending whether we have a MCE,CE, or GCE, is a mixed state of pure states, the microstates of the system. What I am trying to understand is the trajectory of each point (microstate) in phase space. The time evolution implies that the microstate changes over time, but what exactly it changes?

Personally I think that the energy of the microstate doesn't change. Then what does change?

• Microstates evolve under the relevant equations of motion, e.g. the many-body Schrödinger equation in QM. This is what we usually don't have the resources to track, and is what statistical mechanics allows us to circumvent. Jan 11, 2022 at 19:59

"The microstate" is basically all the information you need to completely specifiy your system. For example when you have $$N$$ particles this would mean you need $$N$$ positions and $$N$$ velocities. The phase space is $$\mathbb R^{2N}$$ which can be incredibly large. For $$N=1$$ you can draw the complete trajectories in phase space but this could hardly be called statistical mechanics. To perform time evolution you evolve the particles under the equations of motion. For classical particles this just means to apply Newton's laws.
Another example is a system of $$N$$ coins which can each be head or tails. The phase space is $$(\mathbb Z_2)^{N}$$ where $$\mathbb Z_2=\{0,1\}$$. At each timestep you flip one coin at random. Now "time evolution" means to take one timestep because time is now discrete.
Macrostates are what you get when you forget all microscopic details. For a microcanonical ensemble you only need $$N,V,E$$ to describe the system. All these 3 variables are fixed. When you say something like NVE-ensemble you mean that all these variables are fixed and can be used to describe the system.