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Heat Conduction

Zhigang Suo's picture

So far we have used the fundamental postulate to study experimental phenomena by following an algorithm. For a given phenomenon, we construct an isolated system with an internal variable. The isolated system has a whole set of quantum states. Associated with each value of the internal variable, the isolated system flips among a subset of the quantum states. The fundamental postulate implies that the internal variable evolves in time, from one value corresponding to a subset of the quantum states to another value corresponding to a subset of a larger number of quantum states. After a long time, the internal variable attains an equilibrium value, corresponding to a subset of the largest number of quantum states.

This algorithm 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 algorithm to model several phenomena:

  • A half bottle of water, where the internal variable is the number of water molecules in the vapor (node/290).
  • Wine and cheese in thermal contact, where the internal variable is the amount of energy in the wine (node/291). This model relates temperature to energy and entropy.
  • A more formal analysis of coexistent phases, where the internal variable is the number of molecules in one of the phases. (node/291).

We now use the same algorithm 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 of temperature is inhomogeneous, energy will flow from a place of high temperature to a place of low temperature. When the field of temperature is homogenous, energy will cease to flow, and the body is in a state of equilibrium. This presentation is essentially the same as that of Fourier, formulated in early 1800s, before thermodynamics was established. His formulation may be pictured by using an analogy. A fluid flows from a place of high altitude to a place of low altitude. Heat flows like a fluid, from a place of high temperature to a place of low temperature.

We will then show that Fourier’s theory is consistent with the two great principles: the fundamental postulate and the conservation of energy. We would like to use this familiar phenomenon to describe the algorithm of modeling that will allow us to formulate theories for other phenomena.

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Tony Rockwell's picture

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.

Zhigang Suo's picture

  1. Complete set of equations.  You are right:  the full list of equations are the same as before.  The calculation of entropy shows that, if we adopt Fourier's law, the second law of thermodynamics is satisfied.  In this derivation, we indeed have used energy conservation.  The situation might be understood in a familiar context.  Let's say we want to solve a set of 2 by 2 algebraic equations.  We can use one equation to simplify another, but we will still have two equations.
  2. Independent internal variables.  A more systematic way to select independent variables is to list all the variables and then list all the constraints.  Then decide which variables can vary independently.  In practice, we can think over this matter in several ways, some physical and some mathematical. 
Zhigang Suo's picture

Dear Tony:  Following your suggestions, I have streamlined the notes.  I am teaching the course on advanced elasticity again, and have been updating other sections of the notes as well.

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