Heat Conduction
So far we have used the fundamental postulate to study experimental phenomena in a particular way. For a given phenomenon, we construct an isolated system with an internal variable. As the isolated system flips from one quantum state to another, the internal variable changes in time. Associated with each value of the internal variable is a subset of the quantum states of the isolated system, and the number of the quantum states in this subset. The fundamental postulate implies that the internal variable evolves in time to increase the number of quantum states in the subset.
This statement leads to an algebraic equation to determine the internal variable in equilibrium, and an ordinary differential equation to evolve the internal variable in time. We have used this procedure to model several phenomena, including
- a half glass of wine (http://imechanica.org/node/290),
- two systems in thermal contact (http://imechanica.org/node/291),
- two types of phase transition (http://imechanica.org/node/4878).
We now wish to study phenomena in which internal variables are represented by time-dependent fields. We do not change our perspective on the fundamental postulate. Rather, we change our way to describe internal variables, from using numbers to using fields. This change results in a change in mathematical tools, from algebraic equations and ordinary differential equations to partial differential equations.
Such a change, however, is not as profound as it may appear. Especially in our own time, fields are routinely discretized into numbers, and the computer solves algebraic equations to determine these numbers in equilibrium, or solves ordinary differential equations to evolve these numbers in time. The computer crunches numbers; it knows no partial differential equations.
We begin with a familiar phenomenon: heat conduction. We will first recall the theory of heat conduction presented in undergraduate textbooks. Inside a body is a time-dependent field of temperature. When the field is inhomogeneous, energy will flow from a place of high temperature to a place of low temperature. When the field is homogenous, energy will cease to flow.
We will then show that the above theory is consistent with the fundamental postulate. We would like to use this familiar phenomenon to describe a procedure that will allow us to formulate theories for other phenomena.
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2 questions on heat conduction lectures
I have two questions stemming from the lectures on heat conduction, which may be trivial in nature, but forgive me: blame it on the fact I am a biologist!
First, I am very impressed that by our success in class in applying the fundamental postulate to a variational statement of total entropy and recovering boundary conditions and Fourier's law. The fact that we stopped there suggests that at that point everything is known.
But, to be completely explicit and proceed to solve for the details of the temperature field, it would seem to me that the next step would be to (as usual) combine fourier’s law and an energy conservation statement to find a pde for the temperature field. But, did we not already use a conservation of energy statement for a material particle in our variational expression of the change in entropy of the composite? That is, it seems like we already used conservation of energy in finding our fourier law, but now the only thing I can see to do is to use it again, and thereby derive the usual pde – but this makes me nervous as I have a vague notion one ought not use the same relation twice in deriving a governing equation.
My second confusion: How do we ensure a good choice of independent variables? In the notes and lecture, you state that a good choice of independent variables is I,q, Q. The defense of your choice of seems to rest on physical intuition –
Once we know the energy added to a material particle from the reservoir and from neighboring material particles, the conservation of energy determines the variation of the energy of the material particle, and the thermodynamic model of the material determines the entropy of and the temperature of the material particle.” (p.6).
This makes sense, but is not the requirement of independence a mathematical one? And if that is so, it ought to be evident from the mathematical structure what an appropriate choice would be. The physical argument makes a lot of sense, but I worry as the physical picture gets more complex it won’t be as easy to see.
Re: 2 questions on heat conduction lectures