Understanding classical stochastic systems

[spaghetti3451]

This is an extract from my lecture notes:

"For classical stochastic systems, w(p,x,t)dpdx = prob. particle is in dpdx.
$w\geq0 \int dp \int dx w(p,x) = 1.$"

1. Can anyone please explain what a classical stochastic system is?
2. Why is there a question of probability in analysing such a system?

 

[mathman]

I can't comment on classical. Stochastic processes are time dependent random processes governed by standard probability laws - probability non-negative and total probability = 1.

http://en.wikipedia.org/wiki/Stochastic_process

 

[Bill_K]

failexam, Stochastic is just a fancy word for random. A stochastic system in physics is one that's governed by statistical mechanics.

The word 'classical' is in there because for a classical system x and p may be specified independently and take on any value, so you can speak of w(p,x) as a continuous function in phase space. For a quantum system only some values of x and p are allowed. For example for a harmonic oscillator, ½(p² + x²) = nħ, and so the distribution in phase space is discontinuous.

 

[atyy]

In classical systems, randomness arises because of lack of experimental control. For example, the experimentalist may not be able to prepare the system in exactly the same state every time. He may only be able to prepare the system in some states with higher probability than others. This is how probability enters classical systems.

In quantum systems, even if the experimentalist has perfect control, probability still enters the physics.

Also, for a classical system, w is always greater than or equal to zero. Also, for a quantum system w is not always greater than or equal to zero, but can be negative. This is because a classical particle can have definite position and momentum. However, a quantum particle with definite position cannot have definite momentum.

 

[PJC01]

I would be happy to provide a response to the content you have shared from your lecture notes.

1. A classical stochastic system refers to a system that exhibits random behavior or variability in its physical properties or behavior. This can include systems such as gas molecules moving in a container, atoms in a crystal lattice, or the movement of particles in a fluid. These systems are often described using statistical mechanics and probability theory.

2. The question of probability arises in analysing classical stochastic systems because these systems are inherently unpredictable due to their random nature. This means that we cannot accurately predict the exact behavior or properties of the system at any given moment. Instead, we can only determine the likelihood or probability of a particular outcome occurring. This is why the equation in your notes includes the term "prob." which stands for probability. By understanding the probability distribution of the system, we can gain insight into its behavior and make predictions about its future state.