## Category Archives: Chaotic attractor

### State functions, state equations, graphs of them and reversibility

Thermodynamic States are all considered to be continuous variables (the fact that Internal Energy (U) is a state variable is half of the first law).

A continuous function of state_function_1 in terms of state_function_2, . . . . state_function_n produces a graph which is an n dimensional surface in n + 1 dimensional space. If this seems rather abstract, we’ll get concrete shortly. Consider the classic calculus 101 function y = x^2. Write it like this

f : R^1 –> R^1
f : x |–> x^2

This does seem a bit stuffy, but the clarity it provides is useful, as you’ll see. R^1 is the set of real numbers. The first line tells you that f goes from the real numbers to the real numbers. The second like gives you what f does to a point in the domain. What about the graph of f? It is the parabola, which lives in the x – y plane, a 2 dimensional space. The graph of f is just a curved line with dimension 1, living in a space one dimension higher (e.g. dimension 2).

Different state functions apply to different physical systems at which point they are called equations of state, with every point on their graph representing a collection of state variables at which the system is at equilibrium (e.g. not changing with time)

The simplest state function comes from the ideal gas law PV = nRT, which was promulgated in 1834 by Claperyon. You may regard it as

T : R^2 –> R^1
T : (P, V) |–> P*V/R == T

This is Temperature (statefunction1) in terms of P (statefunction2) and V (statefunction3). What is its graph — something 2 dimensional living in 3 dimensional space — e. g. a surface.

If you’ve studied PChem, you’ve probably met the Carnot cycle. Here’s a link https://en.wikipedia.org/wiki/Carnot_cycle.  It is represented by  a bunch of curved lines in the PV plane, but each line in the diagramreally represents a line on the 3 dimensional graph of T. You can think of this like a topographic map of a mountain, but not quite. The top and bottom lines represent constant temperature (altitude) but the (semi)vertical lines are paths up and down the mountain. Just looking at the flat PV diagram is pretty misleading.

Any combination of P, V, T not satisfying PV = RT is not on the surface, and is not in equilibrium.  You won’t see any of them on the diagram of the PV plane, which is why it’s so misleading.

P, V and T will change so they approach the surface (either by minimizing internal energy or maximizing entropy or a combination of both — these are the driving forces of Dill’s book — Molecular Driving Forces.

The definition of surface given above is quite general and applies to more complicated situations — which is why I went to the trouble to go through it. For instance, in some systems Internal Energy (U) is a function of 3 variables Entropy (S), Volume (V) and the number of molecules (N). This is a 3 dimensional surface living in 4 dimensions. It’s just as much of a surface as that for T in terms of P and V, but I can’t visualize it (perhaps you can) Note also that when you go to higher magnification N is not a continuous variable, any more than concentration is.

Any point on the surface can be reached reversibly from any other — what does reversibility actually mean?

Berry Physical Chemistry 2nd Ed 2000 p. 377. Reversibility of changes in equilibrium means 3 things.

l. The change occurs almost infinitesmally slowly (a very large class of real processes have work and heat values very close to reversible processes)

2. Changes remain infinitesmally close to equilibrium (e.g. they stay on the surface. At equilibrium, thermodynamic variables still fluctuate. If movement on the surface is slow enough that the thermodynamic variables are within 1 standard deviation of the average values of the thermodynamic state variables, no observation can show that the stat eof the system has changed

3. Intensive variables corresponding to work being done (e.g. pressure, surface tension, voltage) are continuous across the boundary of the system on which work is being done.

Objects off the surface aren’t in equilibrium and maximization of entropy or minimization of internal energy drive them toward the surface. This implies that the surface is is an attractor. Now that chaos is well known, are there thermodynamic attractors — I’ve written Dill to ask about this.

Hopefully this will be helpful to some of you. Putting it together was to me. As always, the best way to learn something is trying to explain it to someone else.