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A Simple Model for Energy Sharing (2013)

Page history last edited by wlosert 10 years ago

Working Content> Thermodynamics and Statistical Physics

 

Energy sharing is at the core of Thermodynamics.  This is  already implied in the word itself, which combines dynamics, ie. movement, with the greek word "thermos", which means hot.  The equations in thermodynamics all deal with how energy is flowing between objects and systems, and how energy is flowing within a system. Here we introduce a simple physical model of energy sharing that can help give you insights into how energy is shared within a system and between systems. 

 

Zooming in at individual atoms of molecules this moving around of energy may seem quite random.  When atoms collide, we have learned that they can exchange energy and momentum.  How much energy is exchanged in a collision really depends on the exact details of a collision - impossible to predict.  Nevertheless, we can predict how energy is shared within a system and between systems!  How is this possible? 

 

This is where statistical physics comes in.  While we do not know what is happening in a single collision, we can predict what may happen on average in many collisions and interactions.

 

Lets introduce a simple model system. 

Our model system is a block of energy that we want to share in a number of "bins" or degrees of freedom.  First we will share the block of energy among the degrees of freedom within a system, later we will look at sharing between systems. 

 

Another approximation we will make is that the energy cannot be cut up into arbitrarily small pieces but comes in small chunks.  Initially this was thought to be an approximation that was mathematically needed, but quantum mechanics (which was developed after thermodynamics and statistical physics) showed that indeed energy cannot be divided into arbitrarily small quantities but comes in small but finite chunks.

 

Lets look at one concrete example.  Consider a block of energy consisting of 19 chunks. The figure on the right shows this block of energy divided among two bins. 

The table on the right shows our initial assumption - we assume that each configuration is equally likely for two bins.   Overall there are 20 different ways to slice up the block of energy - one more option than chunks of energy since the amount of energy in bin two, for example can be anywhere between 0 and 19.

 

More generally speaking, if we have m chunks of energy, there are m+1 ways the two bins can share that energy.

 

 

Now lets consider that a third bin is also present, and we distribute the 19 chunks of energy among the three bins.  There are 20 different ways that this third bin can have ZERO energy.  But if the third bin has one chunk of energy, the other two bins only have 18 chunks to share.  There are 19 different ways to share the 18 chunks between bins 1 and 2, so the second row in the table on the right has a "19" in it.  In a similar way we can fill in the rest of the table. 

 

Each of the 20 options for bin 3 now have a large number of options for bin 1 and 2!  This leads to a strong increase in the total number of scenarios:  We now get 210 scenarios simply for sharing 19 chunks of energy among 3 bins!

 

Again, we can consider m chunks of energy.  Now, there are about m2/2 scenarios to distribute the m chunks of energy among three bins.


 

You are starting to see that the number of scenarios increases rapidly.  Indeed the number of scenarios or the number of ways the energy could be distributed is important. 

Lets be more technical and call the scenarios "microstates" and call the number of microstates N.  In thecase shown in the table above we then have N(19)=210 microstates; or more generally N(m)=m2/2.  Each microstate represents one way to distribute a fixed number of chunks of energy into all available bins. 

 

Pushing this model further, we can show that for four bins, the number of microstates N becomes very large: N(m)=m3/(3*2).  This could also be written as N(m)=m3/3!  (the exclamation mark indicates "factorial" i.e. 5! is 5*4*3*2*1 and 3! is 3*2*1

 

As a next step lets consider distributing the energy into n different bins.  This yields N(n,m)=m(n-1)/(n-1)! scenarios.   We can see just how quickly the number of microstates increases by plotting N (n,m) as a function of  the number of energy chunks m. 

 

We can also take the logarithm of N and call it  Entropy or S.  S(n,m)=log(N(n,m)

 

Note that entropy depends on the amount of energy in the system.  As you see on the right, the number of microstates increases quickly with the amount of energy available.  The entropy also increases with increasing energy, the more energy you have to distribute, the more ways you can distribute it.


 

 

That is it for the model!   Now it is time to translate this to thermodynamic variables and look at consequences. 

 

N:  microstates

m:  chunks of energy in the system  -> U

n:  number of "bins" where energy can be stored -> degrees of freedom

 

What we learn from the model:

 

1) The number of microstates increases rapidly with energy in the system, and with the number of bins where energy is stored.  This can be seen in the graphs  N(n,m) and S(n,m) above.

 

2) The slope of the graph S(n,m) vs m has meaning!  The inverse of the slope is temperature!  This is a direct consequence of the equation

 

3) the probability distribution of energy is a Boltzmann distribution.  Looking back for example at three bins holding 19 chunks of energy, we can see that among the 210 scenarios there are more scenarios with zero energy for bin 3 than scenarios and only one scenario where the third bin has all 19 chunks of energy.   In other words, having no chunks of energy is the most likely state for bin 3.

 

For a larger number of bins this becomes more striking.  Lets look at 11 bins.  The number of microstates where bin 11 has no energy is huge, and the odds of having a certain energy in bin 11 increases with increasing energy.  This is shown in the figure on the right.  For large numbers of chunks of energy we find that this curve is a Boltzmann distribution.  On the right you see that even for 11 bins and 40 chunks of energy the curve is almost a Boltzmann distribution.  


 

 

 

4) Two systems equilibrate to the same temperature.  Why would this be the temperature of the system?  Clearly, the temperature increases if increase the amount of energy we put into the system, i.e. the slope of the S(m,n) decreases with increasing m.  But temperature has a special meaning, two objects in thermal contact that exchange energy will reach an equilibrium where both have the same temperature, i.e. the same slope of their entropy vs energy curve? 

 

To see why this is the case, lets consider a fixed amount of chunks of energy that we have to distribute among TWO systems.  The more chunks we give to system A, the more microstates it will have, i.e. the more ways it will have to distribute that energy.  At the same time system B will have less chunks of energy, and so get fewer microstates the more energy we take away from it. 

 

Here is where we have to introduce an important idea again:  entropy is ADDITIVE, i.e. the entropy of the overall system consisting of A and B is the SUM of the entropies. 

 

Now we are ready to think about what state the system will be found it - it is equally likely to be in any microstate, so it will most likely be found at the energy that has the most microstates, i.e. the peak entropy.  In other words the system will evole such that entropy is maximized. 

 

The sum of entropies is a peak at the point where the slopes of the two entropy curves are equal in magnitude and opposite in sign, i.e. the amount of microstates gained for system A when we add one more chunk of energy will be exactly equal to the amount of microstates lost to system B when it looses one chunk of energy.  In other words, at the peak of the total S curve the temperatures of both A and B are equal.
 

 

One important final note:  We started this discussion with a simple model that energy is equally likely to be shared when to bins exchange energy.  It turns out that this assumption is not necessary for any of the conclusions we have drawn.  The same results 1)-4) hold more generally for systems that share energy freely among its degrees of freedom. 

 

 

 

 

Tom Antonsen, Dave Buehrle, and Wolfgang Losert 2/2014

 

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