Entropy inequality via Gibbs states

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Authors

Wlodzimierz Holsztynski

            ENTROPY INEQUALITY SPANS A PATH OF GIBBS STATES

   Let        X  be a finite set.  Let  m : X –> R  be a “probabilistic
   measure” on points, i.e. let  m(x)  be non-negative for every  x  in X,
   and let  SUM(m) = Sum( m(x) | x in X) = 1  (geometrically speaking
   m  is a point of the simplex which has  X  as its set of vertices).
   Then the Shannon entropy is, by definition,

        s(m)  =  (-1) * Sum( m(x)*log(m(x)) | x in X )

             =  I(-log o m)

   where  I  is the integral w.r. to m,  and  “o”  is the symbol of the
   composition of function.

   Also  0*log(0)  is assumed to be 0        (hence entropy        s  is a continuous
   function of        m).

   Let        u : X –> R be the constant function:  u(x) = 1/|X|.
   Then         u  is a probabilistic “measure” in X, we can call it uniform.
   A well known elementary but fundamental inequality states:

        s(m) <= s(u) = log|X|

   for every “probabilistic measure”  m  in X.

   We will prove this inequality by connecting measures  m  and  u
   by an entire path of measures in  X  for which entropy will
   increase smoothly.  The intermediate states (measures) will
   be the Gibbs states obtained from  m  by increasing temperature
   (so to speak).  It’s only natural that higher temperature
   leads to higher entropy  (as we see it on this net all the time).

   Gibbs states for finite systems
   ——————————-

   Consider a finite set  X  and an arbitrary real function

                        U : X –> R;

   X  is called the space of configurations, and  U  is called
   the energy function.  We also define the “partition function”
   Z = Z(T),  where  T > 0  is called temperature:

        Z  =  Sum( exp(-b*U(x)) | x in X )

   where  b = 1/T  (sorry for that :-).

   Finally, the Gibbs state  m_T  at temperature  T  is a “measure”
   given by formula:

                m_T  =        (exp o (-b*U)) / Z

   i.e.

             m_T(x)  =        exp(-b*U(x)) / Z               for every  x in X.

   Obviously,  SUM(m_T) = 1  hence  m_T  is probabilistic.

   When temperature rises then        m_T  converges to the uniform
   state  u:

        lim(m_T(x) |  T –> infinity)  =  1/|X|.

   For the sake of completeness let’s add that:
                                     _
                                    |        0  if  U(x) is not minimal
        lim(m_T(x) | T –> +0)        =  <
                                    |        1/|X_min|  if  U(x) is minimal
                                     –

   where  X_min is the set of all  x  in  X  for which
   energy  U(x)  is minimal in X.

           *-=-=-=-=-*-=-=-=-=-*-=-=-=-=-*-=-=-=-=-*-=-=-=-=-*

   Notation:

        given  f,g: X –> R  let

                <f,g>  =  Sum( f(x)*g(x) | x in X)

        and

                I_T(f) =  <f,m_T>

   (hence, for entropy,  s(m_T) = -I_T(log o m_T).

           *-=-=-=-=-*-=-=-=-=-*-=-=-=-=-*-=-=-=-=-*-=-=-=-=-*

   It’s easy to see that:

        dZ/db           =  – < U, exp(-b*U) >

        d(logZ)/db = – I_T(U)

        dm_T/db    = – (U – I_T(U)) * m_T

   Furthermore, the Shannon entropy of a Gibbs state is easily
   seen to be:

        s(m_T)        =  log(Z) + b*I_T(U)

   Thus it’s once again easy (apply Lagrange inequality) that:

   “MAIN THEOREM” :   ds(m_T)/db  =  b*((I_T(U))^2 – I_T(U^2)) <= 0
   for every  T>0.

   Corollary 1.  d(s(m_T))/dT >= 0   for every        T>0.

   Corollary 2.  s(m_Q) <= s(m_T)   whenever  0 < Q <= T.

   Corollary 3.  log(|X_min|) <= s(m_T) <= log(|X|)  for every        T>0.

   Shannon entropy for general finite probabilistic spaces
   ——————————————————-

   Let        n : X –> R be an arbitrary positive “measure”, i.e.
   let        n(x) > 0  for every  x        in  X,        and let  SUM(n) = 1.
   Define  U : X –> R        as follows:

        U(x) = -log(n(x))    for every        x in X.

   Then  the “measure”  n  and the Gibbs state  m_1  (at
   temperature 1) coincide:  m_1 = n.        Thus we immediately
   obtain from        Corollary 3  above that:

   THEOREM.  s(n) <= log(|X|)  for every “probabilistic measure”
   n  in  X.

   (For all of them, not just for the strictly positive ones.)

        
   An attempt to demystify Gibbs States
   ————————————

   It is enlighting and interesting to have the notions of physics
   help us to do pure mathematics.  Such things happen at least
   since Archimedes.  We get more motivated and get a better understanding.
   Or do we?  Would the language of physics obscure our, simple after all,
   inequality about entropy.

   Indeed, we do not need physics to connect any given “probabilistic
   measure” with the uniform one by a smooth path.  Let

        Z_T =  SUM( m^b )  =  Sum( (m(x))^b | x in X )

   for every  T>0  and  b = 1/T.  Then  m_T : X –> R  defined by:

         m_T  =  m^b/Z_T

   is a “probabilistic measure”.  Obviously,  m_1 = m  and  lim m_T
   is the uniform measure when  T –> infinity.

   No Gibbs states.  Actually these  m_T  measures are the same
   Gibbs states as earlier but we don’t need to know it.  Hence so far
   so pure,  as an “ultramathematician” would say.

   Now let’s try again to prove that the entropy  s(m_T)  increases
   when the temperature  T  does:

      s(m_T)  =  log(Z_T)  –  b * SUM( m_T * (log o m) )

             =  log(Z_T)  –  b * I_T(log o m)

   etc.  We can continue just as we did earlier.  But if we don’t
   recognize the role played by   U = -log o m   then, well, then
   we will understand less than before,  e.g.  our simple proof will
   look  ad hoc  and  like a mess.  Hence:

   CONCLUSION. The formalism of the Gibbs states adds to our
   understanding of the notion of entropy in general (outside
   of physics, in the realm of pure mathematics).

        
Enjoy (or hate 🙂

        Wlodek

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