445 PHYS Lecture 8 - Interacting systems Lecture 8 - Interacting systems What's Important: equilibrium and reversibility systems in interaction Text: Reif
8-1
Constraints and
(Reif, Sec. 3.1; not covered in class)
Suppose that two systems are brought in contact, and a constraint that isolated them is removed. How do we describe theoretically what happens? For example, consider what happens when a thermal barrier between two gases is removed, but the volume of their container is not altered.
Remove the barrier Once the barrier has been removed, the number of accessible states must be the same, or larger: f i. That is, removing a constraint cannot decrease the number of accessible states. As the probability of the system being in a particular state is proportional to , then the probability of a system returning to its initial state is i / f. Reversible and irreversible processes (Reif, Sec. 3.2; not covered in class) If the number of accessible states doesn't change during a process, that is f = i, then equilibrium can be maintained during the process and it is reversible. If the number of accessible states increases during a process, f i, then the process is irreversible. For example, if a gas expands to occupy a larger volume by removing a barrier, then it will not go back to its original state if the barrier is reinserted:
2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.
PHYS 445 Lecture 8 - Interacting systems
8-2
Remove the barrier Systems in interaction (thermal contact) Note: the following argument uses discrete energies, but goes over to continuous energies without modification) Next, consider what happens when we place two systems in thermal contact, but not mechanical contact (i.e., systems can exchange energy, but not volume or material).
A
A'
Ao
Two systems A and A' combine to form Ao, which is isolated. Because Ao is isolated, the total energy of Ao is constant, even as the subsystems exchange energy. Denoting the energies of A, A' and Ao as E, E' and Eo, then E + E' = Eo = constant (Ao is isolated) Denote the number of accessible states in each system as (E), '(E') and o(Eo). What is the probability P(E) of system A having energy E? In the isolated system A before contact, it is (E ) (E) PA(E) = = (8.1)
TOT
What about the combined system? Call PAo(E) the probability of system A having energy E once the systems are in contact. Then 0 (E) 0 PA (E) = (8.2) 0
TOT
with
0 TOT
=
E
0
(E )
(8.3)
What about o(E)? For every state in the collection (E), a total of 1 x '(E') combined states can be formed (where E' = Eo-E). Hence, for a specific energy division E and E', there must be a total of (E) '(E') states. We sum over all of these combinations to obtain oTOT.
2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.
PHYS 445 Lecture 8 - Interacting systems
8-3
Numerical example Let systems 1 and 2 have allowed energies E = 1,2 and E' = 1,2, with accessible states according to # of states E # of states E' Combined states E' = 1 E= 1 16 2 32 2 1 8 1 2 4 8 4 2 2 2 total energy E' = E= 1 2 Eo 1 2 3 2 3 4
Combined states as a function of Eo: # of states 16 32+4 8 Eo 2 3 4 o for a total of TOT = 60 states. Fractional distribution of combined fraction states of states 0.27 0.6 0.13 Eo 2 3 4 Now, let's see the effects of constraints. If we ask, what are the number of states of the combined system o as a function of the energy E of the subsystem, we would find (add rows in the "combined states" table): o (E) 20 40 E 1 2 (no constraint on Eo) As expected, the fractional distribution of combined states is the same, as a function of E, as the uncombined subsystem. The same would be true if we calculated o as a function of E' (by adding columns). BUT, if we place a constraint on Eo, the fractional distributions of the subsystems changes. Say we demand Eo = 3. Then the number of states o as a function of E of the subsystem changes to (cross diagonal in the table) o (E) 4 32 E 1 2 (constraint Eo = 3). Hence, once the systems are put in thermal contact, and constraints are placed on the energy of the combined system, then the probability distribution in a subsystem may be changed. The value of E to which subsystem A changes once the systems are placed in thermal contact (and the total energy is constrained!) can be calculated from the distribution of total states Po, as a function of E. Let's move away from our numerical example, and consider the problem in general.
2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.
PHYS 445 Lecture 8 - Interacting systems
8-4
The number of states
of A as a function of E generally increases rapidly with E:
E The number of states ' of A' generally increases rapidly with E', and hence generally decreases rapidly with E = Eo-E' '
E Hence, the probability P oA for system A, once its in contact with A' and the total energy is constrained, looks like P oA
E ~ E ~ The most likely energy of A is now at E , which can be found by the usual calculus procedure of setting a derivative equal to zero. In this case, we use the natural logarithm of P oA: P oA is a maximum when lnPA0(E) =0 E
(ln E
E E
(E ) + ln '(E ') - ln E' E'
0 TOT
(E )) = 0
(8.4) (
o TOT
does not depend on E)
ln (E ) - ln (E ) =
ln '(E') - 0 = 0 ln '(E ')
Note that a minus sign arises in changing from / E to / E'.
2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.
PHYS 445 Lecture 8 - Interacting systems
8-5
~ This equation by itself does not specify E : rather, it gives a family of E-E' values for which the slopes of the two distributions agree. To determine which member of the family is the solution, one must impose the condition E' = Eo - E. At equilibrium, ln / E is the same for both systems in thermal contact, according to Eq. (8.4). Thus, ln / E is a very useful quantity, which we define as : ln 1 (8.5) E kBT Units: -1 has units of energy kB is a constant, with units of energy T is a dimensionless parameter, identified with temperature. As will be shown, ln is also a useful quantity. Using the same constant kB, we define another quantity called the entropy S kBln . (8.6) Hence, the temperature can alternately be extracted from S T= E At equilibrium, maximizing P implies that S + S' is a maximum T = T' (systems in thermal contact have the same temperature).
(8.7)
2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.