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A general definition of entropy was formulated by Boltzmann in 1872. It is expressed in terms of "coarse-graining volume" in the phase space, which amalgamates the positions and momenta of all particles in a system into one point (Figure 01f). The relentless increase toward higher entropy until reaching its maximum (i.e., in a state of thermal equilibrium) is related to the fact that the evolution of the phase point is more favorable toward the | ||

## Figure 01f Phase Space [view large image] |
## Figure 01g Phase Space Evolution |
larger "coarse-graining volume" (Figure 01g). |

- More details of the definition and its implication are presented in the followings:
- Configuration Space - It is a space consists of all the 3-dimensional spatial coordinates of N particles (N = 4 in Figure 01f, represented by the blue arrows) with all the 3N coordinate axes orthogonal (perpendicular) to each others. The horizontal axis for the phase space in Figure 01f is a much simplified visual aid for the 3N configuration space. At 300
^{o}K and standard atmospheric pressure of 101 kpa, the number of gas molecules N in a cube of 10 cm would be about 3x10^{22}. - Momentum Space - In addition to the position of the particle, each one needs at least three more numbers to specify its state, namely the three components of its momentum (red arrow in Figure 01f). Similar to the configuration space, the momentum space is made up by 3N orthogonal axes representing the momenta of N particles. At 300
^{o}K and standard atmospheric pressure of 101 kpa and assuming the gas molecules to be hydrogen atoms with mass m = 1.67x10^{-24}gm, the root-mean-square velocity of the particles v_{rms}= (3kT/m)^{1/2}~ 3x10^{4}cm/sec., the corresponding momentum p = mv_{rms}= (2mE)^{1/2}~ 4.5x10^{-20}erg-sec/cm (or E ~ 10^{-16}erg ~ 6x10^{-5}ev). The size of the momentum space for each particle can be estimated from a range below and above the rms value such that about 0.1% probabilities toward the tail ends are excluded. - Phase Space - It is the orthogonal combination of the configuration and momentum spaces having altogether 6N dimensions as shown in Figure 01f. The dimensions are often referred to as the degrees of freedom. The phase space volume W is:

W = {[^{3N/2}(2mE)^{3N/2}V^{N}]/[(N!(3N/2)]}(E/E), where

p = (2mE)^{1/2}(E/2E) is the range of momentum,

2^{3N/2}(2mE)^{(3N-1)/2}comes from integrating up to the energy E = p^{2}/2m,

V is the spatial volume containing the particles,

N! and (3N/2) are for removing the degeneracy related to the permutation symmetry of identical particles. (3N/2) is the Gamma function identical to (3N/2)! if the argument is an integer. - Partition Function - It is the the number of microscopic states within the energy shell E of the phase space. The Planck's constant h = 6.625x10
^{-27}erg-sec from the uncertainty relation px ~ h in quantum theory is conveniently taken as the basic unit (minimum size) of the microscopic states. Thus the partition function Z is just:

Z = W/h^{3N}= {[(^{1/2}(2mE)^{1/2}V^{1/3})/h]^{3N}/[N!(3N/2)]}(E/E) ~ {(10^{8})^{3N}/[N!(3N/2)]}(E/E)

where the numerical value 10^{8}is computed from the previous assumptions for the size of the container and p. It shows that the number of microscopic states available is enormous in the order of 10^{24}even for a system of just one particle (N = 1).

- Entropy - Boltzmann's definition of entropy S is:

S = k ln(Z)

where k = 1.38x10 ^{-16}erg/^{o}K is the Boltzmann constant. It is immediately clear that entropy would increase by adding # of particles N, energy E, or volume V as shown in Figure 01d (the internal degrees of freedom are not considered here). Since Z depends on the parameters in power of 3N, it varies by huge amount with a relatively small change in this parameter.#### Figure 01h Entropy and Randomness

_{}Figure 01h illustrates the connection of entropy with randomness once the support of orderliness is removed.

- Coarse-graining Region - Each of the sub-volume w in the phase space (Figure 01i) is characterized by some macroscopic properties such as
temperature, pressure, density, color, chemical composition etc. which are some sort of averages out of a humonogous number of microscopic states - the multiplicity (see example in Figure 01j, = Z) of positions and momenta. Its number of neighbors goes up drastically with increasing dimension, e.g., typically 6 in the 2 dimensional case, 14 in 3 dimensions, ... As illustrated by formula above, the various w sub-volumes tend to differ in size by absolutely enormous factors. #### Figure 01i Coarse-graining Regions [view large image]

#### Figure 01j Multiplicity

[view large image]

- Second Law of Thermodynamics - The evolutionary path of a phase point in the phase space is indicated by a curve as shown in Figures 01g, 01i. Although time and hence rate of change is absent in the picture, the direction of evolution is represented by an arrow (see arrow of time, Figure 01k). The path is determined by physical law such as the N-body Newtonian equation of motion, it has a higher probability of moving into another w sub-volume with larger size and hence higher entropy - the basic conception of the Second Law of Thermodynamics. The appearance of randomness is the manifestation of so many different microscopic states available for the same macroscopic state. The system

reaches thermal equilibrium when the phase point enters the largest sub-volume and keeps wandering around inside (Figures 01h, 01i). Note that there is a certain probability of going into a smaller w, but the probability goes down rapidly with decreasing sub-volume size. #### Figure 01k Arrow of Time [view large image]

the entropy of the black hole is reduced to at most three bits : mass, angular momentum (if rotating), and charge (if carrying any). See "Cosmic Entropy Evolution".- Evolution - Evolution of the phase point is governed by physical forces as well as the size of the coarse-grains. For examples, a refrigerator would restrict the point to a range of space and momentum corresponding to certain temperature T and fixed volume V; air molecules of the atmospher are confined to a layer of about 100 km by Earth's gravity, ...
#### Figure 01l Entropy Evolution

_{}Figure 01l,a shows the evolution of entropy in free space with diffusion of particles. In Figure 01l,b the system is collapsing to black hole by the pull of gravity, the increase in entropy is via the Hawking radiation. Meanwhile

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