## Tuesday, September 22, 2015

### Entropy

• The entropies of gases are much larger than those of liquids, which are larger than those of solids (columns 1, 3, and 4).
• This can be predicted from equation dS=dQ/T: heat must be put into substances to convert them from solid to liquid or liquid to gas. Therefore, q and DS are both positive and the liquid or gas has more entropy than the solid or liquid. On the nanoscale level, the atoms in solids are constrained to one position; they can only vibrate around that position. The atoms in liquids are still close together but they are free to move around with respect to each other, so they are more disordered. The atoms in gases are far apart from each other, so they are much more disordered than either liquids or solids.
• Entropies of large, complicated molecules are greater than those of smaller, simpler molecules (column 2).
Large, complicated molecules have more disorder because of the greater number of ways they can move around in three-dimensional space.
• Entropies of ionic solids are larger when the bonds within them are weaker (columns 3 and 4).
If you think of ionic bonds as springs, a stronger bond will hold the ions in place more than a weaker bond. Therefore, the stronger bond will cause less disorder and less entropy. Two more patterns emerge from considering the implications of the first three.
• Entropy usually increases when a liquid or solid dissolves in a solvent.
Before mixing, the solute and solvent are completely separated from each other. After mixing, they are completely interspersed within each other. Thus, the entropy increases.
• Entropy usually decreases when a gas dissolves in a liquid or solid.

====================

Shannon Entropy

H = + sum [ (n/N)*log2(1/(n/N)) ] = - sum[ (n/N)*log2(n/N) ]  (it will come out positive)

Where "n" is the number of times a distinct symbol occurs in a message of length N symbols.   Probability of symbols beforehand is assumed to be equal.

If the symbols in a message occur with equal probability in that message, Shannon entropy of that message is simply log2(number of symbols).  Otherwise, the entropy is lower, assuming the same number of symbols are used

Example messages and their Shannon Entropy:

0 or 1 or 0000 or 11111, H = 0
01 or 010101 or 00001111 or 01101010 (four 0's and four 1's), then H=1
abc or abcabcabc or aabbcc or acbbca, then H=1.58
abcd or abcd or abcdabcdabcdabcd or aaabbbcccddd, then  H=2
Note that if abcd in the above is encoded with a=00, b=01, c=10, d=11 then H=1.
.

Notes:
1) Shannon Entropy H is bits per symbol.
2) A message that repeats is the same H as if it was sent only once.
3) Physical entropy can find parallels with N*H, or   but it is not Shannon Entropy H.
4) Repeating symbols in a message of the same length lowers Shannon Entropy (more surprises due to an expectation of randomness).

5) A source sending typical English will have lower shannon entropy (more surprises) due to the repetition of symbols.  To cancel this fake surprise, aka to make this fake lower entropy to be more objective, we have to do something like maybe divide by the Shannon entropy we expected. Or rather, divide each n/N we encounter by the n/N we expected, our expected n/N's should add up to N.  So I have H=sum(n/nexpected)*log2(nexpected/n).

H / (number of symbols in alphabet) may have some use.  Might be same as H/(word length) aka metric entropy.
==========
An excellent intro to quantum entropy in a  (Einstein) solid can be found here:
http://hyperphysics.phy-astr.gsu.edu/hbase/therm/einsol.html

Note: the total energy macrostate is "3" in the example, and the possible ways of getting "3" form 4 oscillators who each have equal and independent probabilities of the energy being 0 to 3 (a "q" microstate value) is
S=k*[ ln((q+N)!)-ln(q!)-ln(N!) ]
Sterling's approximation for large N and for a system q=N gives
S=k*[(q+N)*ln(q+N) - q - N - q*ln(q) + q - N*ln(N) + N ]
S= k*[ 2N*( ln(2)+ln(N) ) - 2N*ln(N) ] = k*2*ln(2)*N = 1.39*k*N

Notice that q and N here are the number of possible symbols (q+1 since zero is an energy state) AND the length of the message (N). Or vice versa, depending on terminology.  Notice that q and N are mathematically the same as far as the entropy is concerned in this simple Einstein solid.  They discuss q>>N but the symbols cold be reversed and the discussion would take a different perspective. HOWEVER, it does not make makes much sense for N>q because q is the sum of energies for each oscillator N. In other words, if there are 1000 possible states of energy and the average state is 500, then q=500*N. If there are 3 states, 0, 1, and 2 "Joules" with equal probability, then q=N.  In this case the physical entropy is S=k*ln(2)*N.

===
thermo state can be specified by one of two cardinal functions, internal energy or entropy and they have a reference basis. U can be function of N,V, and S, and S of U,N,and V.  U=sum of pi's of each Ei of the microstates.  Mass, entropy, or volume added to a system will change its U.  E added to a system will change N,V,S from which U can be calculated. E is added it is also equal to U increase. E can include Q.For an ideal gas:  U=constant*e^S/(cN) * (N/V)^(R/c) * N where c is heat capacity (J/K, i.e. dQ needed for dT, i.e. a percentage J/J since T is kinetic energy). This comes from wiki on internal energy.  Rearranging:

S=1/constant * c*N*ln[U/N*(V/N)^(R/c)]

S=N/constant *[ c*ln(U/N)  + R*ln(V/N) ]

This must come out positive, i.e.,   U and V sort of greater than N.

See wiki on Sackur-Tetrode equation.

Valid only for V/N >> 3E36.  (for oxygen gas?)  so maybe not really useful

R=kb*avagrado, k is J/K/moles heat per temp per particle, so R is also a "unitless" dQ/dT (J/J) like c.  N thereby becomes just a count of "bit spaces" (nits). U/N and V/N are  like the possiblities in 8 bits is 256 = 2^8, so log2(256) = 8.  8 bits memory can store 256 possibilities.  So U/N and V/N are possibilities per memory location.  N = number of memory locations (like bits).  c and R are "base adjustments", getting energy and volume in the same entropy units.

U of a closed system like the Earth is dU=dQ+dW=TdS-pdV. (Q is received, W is done). Add u*dN to get general internal energy where u is potential energy for each N added.

Gibbs free energy changes do not allow S to change.