Understanding classical stochastic systems

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Discussion Overview

The discussion revolves around the concept of classical stochastic systems, exploring their definitions, the role of probability in their analysis, and distinctions between classical and quantum systems. It includes theoretical considerations and conceptual clarifications.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant requests an explanation of what constitutes a classical stochastic system and the significance of probability in analyzing such systems.
  • Another participant describes stochastic processes as time-dependent random processes governed by standard probability laws, emphasizing non-negativity and total probability equaling one.
  • A different participant explains that "stochastic" refers to randomness and that classical systems allow independent specification of position and momentum, contrasting this with quantum systems where only certain values are permissible.
  • It is noted that randomness in classical systems can arise from a lack of experimental control, leading to varying probabilities in state preparation, while quantum systems inherently involve probability regardless of control.
  • Discussion includes the assertion that the weight function w is non-negative in classical systems, whereas in quantum systems it can be negative, highlighting differences in the nature of particles in each context.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of classical versus quantum stochastic systems, indicating that multiple competing perspectives remain without a clear consensus.

Contextual Notes

Some assumptions about the definitions of classical and quantum systems may not be fully articulated, and the discussion does not resolve the complexities of how probability functions within these frameworks.

spaghetti3451
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This is an extract from my lecture notes:

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

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

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