![]() We have already discussed the stability with respect to energy fluctuations in the preceding Chapter and postpone further discussion of the necessary criteria until the next Chapter. In the situation where and stand for externally imposed temperature and pressure and we are considering fluctuations of energy and volume, it is easy to see that such fluctuations will be subject to a restoring force if the curvature of the surface is positive throughout. Lastly, it is convenient to graph extensive quantities per particle. Thus, the positions of the minima are not affected. Note we must use the entropy as a function of energy, volume, and particle number for both and ! In the graph, we set and and note that the precise choice of, , and affects only the vertical, but not the lateral position of the graph. Obtained by taking the dependence derived in the last Chapter:Īnd substituting and. As an example, we show the surface for the ideal gas: 2 expresses the formal condition for the location of the minimum on the surface. On the other hand, given specific values of externally imposed temperature and pressure, the equilibrium energy and volume correspond to the minimum of the function, where Eq. 2: On the one hand, they tell us the values of temperature and pressure that are needed to achieve specific equilibrium values for the energy and volume. ![]() Just as for the thermodynamic potential, there are two ways to interpret Eq. The function, then, represents the sought thermodynamic potential governing the fluctuations of both the energy and volume when the temperature and pressure are externally imposed. 4! To be clear, the parameters and are kept constant during the minimization as well. 3 results just as well from minimization with respect to of the following functionĪt the same time, we can readily convince ourselves that minimizing this function with respect to also yields Eq. 3 resulted from minimizing the function with respect to. Note we have explicitly indicated that the particle number is being kept constant, which will be of use later one. We are mindful that the above equation pertains to the equilibrium values of all the quantities. Perhaps the easiest way to determine the pertinent thermodynamic potential is to first write down the energy conservation law in a form that brings out the energy and volume dependence of the entropy: In an open container, clearly one can only control the pressure. For instance, most liquids do not fill their container fully, and so even if the container is completely rigid (which is an idealization) and fully sealed, the liquid will occupy only a portion of the container, the rest occupied by its vapor or the corresponding crystal. Instead, we can only be sure of the value of the pressure. In many, of not most cases of interest, the volume of the system cannot be rigidly controlled. Next we determine the thermodynamic potential that controls the equilibration of the volume, in addition to controlling the equilibrium value of the energy. drivers/char/random.c in the linux-2.6 tree.8 Gibbs Free Energy, Entropy of Mixing, Enthalpy, Chemical Potential, Gibbs-Duhem it's actually pretty interesting and pretty well commented - you might want to study it. Of course, if you use the "urandom" device, the blocking doesn't happen and the pool simply keeps getting hashed and mixed to produce more bytes, which turns it into a PRNG instead of an RNG.Īnyway. At the same time, there's a bunch of accounting going on - each time some entropy is added to the pool, an estimate of the number of bits of entropy it's worth is added to the account, and each time some bytes are extracted from the pool, that number is subtracted, and the random device will block (waiting on more external entropy) if the account would go below zero. When it's desired to get some randomness out of the pool, the entire pool is hashed with SHA-1 to get the output, then the pool is mixed again (and actually there's some more hashing, folding, and mutilating going on to make sure that reversing the process is about as hard as reversing SHA-1). It takes several variously-trusted entropy sources and mixes them into an "entropy pool" using a polynomial function - for each new byte of entropy that comes in, it's xor'd into the pool, and then the entire pool is stirred with the mixing function. ![]() ![]() Since you mention /dev/random - on Linux at least, /dev/random is fed by an algorithm that does very much what you're describing. ![]()
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