11.3.1: Partition Model

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Page ID: 52430

Paul Penfield, Jr.
Massachusetts Institute of Technology via MIT OpenCourseWare

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Let us model the system and its environment (for the moment consider only one such environment) as parts of the universe that each have their own set of possible states, and which can be isolated from each other or can be in contact. That is, the system, considered apart from its environment, has states which, at least in principle, can be described. Each has an energy associated with it, and perhaps other physical properties as well. This description is separate from the determination of which state is actually occupied—that determination is made using the Principle of Maximum Entropy.

We also assume that the environment has its own set of states, each with its own energy and possibly other physical properties. Again this description of the states is independent of which states are actually occupied.

Our model for the interaction between these two (or what is equivalent, our model for the way the total combination is partitioned into the system and the environment) is that the combination has states each of which consists of one state from the environment and one from the system. Thus, for example, if the system has four states (as our simple two-dipole model does) and the environment has 1000 states, then the combination would have 4000 states. Each state of the combination corresponds to exactly one state of the system and exactly one state of the environment.

We need a notation to keep things straight. We will use the index \(i\) for the system and the index \(j\) for the environment. Then we can denote the states of the total combination using both \(i\) and \(j\), in the form \(i\),\(j\) just like the notation for joint probability (which is exactly what it is). A sum over the states of the total combination is then a sum over both \(i\) and \(j\).

We will assume that it is possible for the system and the environment to be isolated from one another (the dipole drawing shows a vertical bar which is supposed to represent a barrier to interaction) and then, at other times, for the two to be interacting. Whether they are isolated or interacting does not affect the states or the physical properties associated with the states, although it may affect the probability of occupancy of the states.