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What is a "microscopic realization" of a system?

The context is statistical mechanics. The microscopic system consists of many atoms (too many to track individually) with an assigned probability density function f(x,y,z,Vx,Vy,Vz,t).

The macroscopic system consists of the atoms taken together, with macroscopic quantities computed as expectations of microscopic quantities.

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  • $\begingroup$ Can you add some context? What's the paragraph preceding and following this sentence? $\endgroup$
    – BMS
    Sep 3, 2014 at 9:03
  • $\begingroup$ Added some context. $\endgroup$
    – jjack
    Sep 3, 2014 at 9:27

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Statistical mechanics relies on a probabilistic understanding of the world and as such one needs to define a probability space. In classical statistical mechanics the probability space consists of a domain which is the set of all possible microstates (that is the position and velocity vectors of all the particles in the system) and a probability measure associated to this space. This probability measure ensures that all microstates incompatible with the constraints on your system (e.g. fixed number of particles, fixed volume etc...) have probability zero. The set of constraints on your system define what is called a statistical ensemble.

In equilibrium classical statistical mechanics, the probability measure is invariant under the law of classical mechanics and is therefore time independent. Traditionally, we tend to equate the operational notion of statistical ensembles and the probability measures they correspond to in equilibrium statistical mechanics and we use essentially four measures/ensembles (microcanonical ,canonical, grand canonical and canonical-isobaric).

Now, the fact that we use probabilities is in essence no different than our use of probabilities when we cast a dice, except that here the dice has many many faces (which correspond to all the microstates one can imagine).

Now, in the same way that when you cast a dice, a realization would be any number between 1 and 6 (for instance 3), then for a thermodynamic system with say fixed $(E,N,V)$, a microscopic realization can be any microstate compatible with these constraints (for instance all the particles in the corner of the box with one particle having all the energy $E$ of the system and the rest of the particles having no motion). You just have to imagine that you have in your hands a god like device that can tell you the instantaneous positions and velocities of all the particles in your system. Each time you perform a measurement with this device, you will be observing a microstate compatible with the statistical ensemble and hence observing a microscopic realization of this ensemble.

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