No-Go Result for the Thermodynamics of Computation. All computation generates heat. Disposing of that heat has traditionally been a majorfactor in the practical design of computers. The thermodynamics of computation seeks to find the minimum heat generation that must arise in computation.
The first step to this minimum is to make the machines smaller. They use less power and generate less heat. Once we are employing molecular-scale devices, are we still faced with some minimimum of heat generation? A modern consensus answers yet.
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It is based on an idea that is appealing, at least initially. The logical specification of the operations computed sets the lower limit.
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Logically irreversible operations, like erasure, are the culprits. The corresponding universal scaling functions are discussed for various boundary conditions and geometries. Direct measurements and applications for colloidal suspensions are discussed in detail.
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The borderlands between elasticity and hydrodynamics lead naturally to a number of problems in elastohydrodynamics. I will discuss some phenomena this rich area: the physics of fluid-infiltrated soft solids, thin film elastohydrodynamics in adhesion and lubrication, and the mathematical description of singularities associated with touchdown. Biology presents an astounding diversity of discrete states or species, that coexist over time-scales much longer than the characteristic times of the underlying degrees of freedom.
Furthermore, these phenomena range from the scale of gene regulatry patterns to cells and to that of populations. On the sub-cellular scale, molecular competition and positive feedback acting on short time scales maintain cells in specialized states, setting the foundation for complexity of multicellular organisms.
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Statistical Mechanics (Spring, ) | The Theoretical Minimum
We have discussed some of the properties of large numbers of intercolliding atoms. The subject is called kinetic theory, a description of matter from the point of view of collisions between the atoms. Fundamentally, we assert that the gross properties of matter should be explainable in terms of the motion of its parts. We limit ourselves for the present to conditions of thermal equilibrium, that is, to a subclass of all the phenomena of nature.
The laws of mechanics which apply just to thermal equilibrium are called statistical mechanics , and in this section we want to become acquainted with some of the central theorems of this subject.
DAY1: TUESDAY 21
That tells us something about the mean square velocities of the atoms. Our objective now is to learn more about the positions of the atoms, to discover how many of them are going to be in different places at thermal equilibrium, and also to go into a little more detail on the distribution of the velocities. Although we have the mean square velocity, we do not know how to answer a question such as how many of them are going three times faster than the root mean square, or how many of them are going one-quarter of the root mean square speed. Or have they all the same speed exactly?
So, these are the two questions that we shall try to answer: How are the molecules distributed in space when there are forces acting on them, and how are they distributed in velocity? It turns out that the two questions are completely independent, and that the distribution of velocities is always the same. The distribution of the velocities of the molecules is independent of the forces, because the collision rates do not depend upon the forces.
Let us begin with an example: the distribution of the molecules in an atmosphere like our own, but without the winds and other kinds of disturbance. Suppose that we have a column of gas extending to a great height, and at thermal equilibrium—unlike our atmosphere, which as we know gets colder as we go up. We could remark that if the temperature differed at different heights, we could demonstrate lack of equilibrium by connecting a rod to some balls at the bottom Fig.
So, ultimately, of course, the temperature becomes the same at all heights in a gravitational field. If the temperature is the same at all heights, the problem is to discover by what law the atmosphere becomes tenuous as we go up. In other words, if we know the number of molecules per unit volume, we know the pressure, and vice versa: they are proportional to each other, since the temperature is constant in this problem. But the pressure is not constant, it must increase as the altitude is reduced, because it has to hold, so to speak, the weight of all the gas above it.
That is the clue by which we may determine how the pressure changes with height. Note that if we have different kinds of molecules with different masses, they go down with different exponentials. The ones which were heavier would decrease with altitude faster than the light ones.
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Therefore we would expect that because oxygen is heavier than nitrogen, as we go higher and higher in an atmosphere with nitrogen and oxygen the proportion of nitrogen would increase. This does not really happen in our own atmosphere, at least at reasonable heights, because there is so much agitation which mixes the gases back together again.
It is not an isothermal atmosphere. Nevertheless, there is a tendency for lighter materials, like hydrogen, to dominate at very great heights in the atmosphere, because the lowest masses continue to exist, while the other exponentials have all died out Fig. Here we note the interesting fact that the numerator in the exponent of Eq.
That may be an accident, i. However, we can show that it is a more general proposition. Suppose that there were some kind of force other than gravity acting on the molecules in a gas.
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For example, the molecules may be charged electrically, and may be acted on by an electric field or another charge that attracts them. Or, because of the mutual attractions of the atoms for each other, or for the wall, or for a solid, or something, there is some force of attraction which varies with position and which acts on all the molecules. Now suppose, for simplicity, that the molecules are all the same, and that the force acts on each individual one, so that the total force on a piece of gas would be simply the number of molecules times the force on each one.
Energy can be generated, or lost by the atoms running around in cyclic paths for which the work done is not zero, and no equilibrium can be maintained at all. Thermal equilibrium cannot exist if the external forces on the atoms are not conservative.
This, then, could tell us the distribution of molecules: Suppose that we had a positive ion in a liquid, attracting negative ions around it, how many of them would be at different distances? If the potential energy is known as a function of distance, then the proportion of them at different distances is given by this law, and so on, through many applications.