Summation formula from statistical mechanics

In summary, the summation formula in statistical mechanics is a tool used to calculate the total energy of a system by summing up the individual energies of all particles. It is derived from the principles of statistical mechanics and uses variables such as the number of particles, energy of each particle, and temperature. This formula is significant in determining the thermodynamic properties of a system and can be applied to a variety of systems. However, the specific form of the formula may vary depending on the system being studied and its assumed behavior.
  • #1
hilbert2
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About proving a summation rule that appears in theory of spin chains.
I ran into this kind of expression for a sum that appears in the theory of 1-dimensional Ising spin chains

##\displaystyle\sum\limits_{m=0}^{N-1}\frac{2(N-1)!}{(N-m-1)!m!}e^{-J(2m-N+1)/kT} = \frac{2e^{2J/kT-J(1-N)/kT}\left(e^{-2J/kT}(1+e^{2J/kT})\right)^N}{1+e^{2J/kT}}##

where the ##k## is the Boltzmann constant, ##J## is strength of magnetic interaction and ##T## is the absolute temperature.

It's quite simple to check that this holds for ##N=1##, ##N=2## and ##N=3##. However, it seems to be a bit more difficult to actually prove by induction than some other sums like

##\displaystyle\sum\limits_{m=1}^{N}m^2 = \frac{N(1+N)(1+2N)}{6}##,

where only the last term of the sum changes when ##N\mapsto N+1##.

If anyone wants to try to prove this as an exercise, be my guest and post it here... It's probably just about applying the binomial theorem.
 
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First we write ##\alpha =\dfrac{J}{kT}## and cancel ##2e^{(N-1)\alpha}## on both sides. Then the equation reads
$$
\sum_{m=0}^{N-1} \binom{N-1}{m}e^{2m\alpha} = \left( 1+e^{2\alpha} \right)^{N-1}
$$
 
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  • #3
Thanks,

so it's just the known formula

##\displaystyle (1+x)^n = \sum\limits_{k=0}^n \binom{n}{k}x^k##

with substitutions ##n=N-1##, ##k=m## and ##x=e^{2\alpha}##.
 
  • #4
hilbert2 said:
Thanks,

so it's just the known formula

##\displaystyle (1+x)^n = \sum\limits_{k=0}^n \binom{n}{k}x^k##

with substitutions ##n=N-1##, ##k=m## and ##x=e^{2\alpha}##.
Yes, although the right hand side is written a bit more complicated than necessary:
$$
2x^{-\frac{n}{2}}(1+x)^n =2e^{-(N-1)\alpha} \cdot (1+e^{2\alpha})^{N-1} = \dfrac{2e^{2\alpha + (N-1)\alpha }\left( e^{-2\alpha} \left( 1+e^{2\alpha} \right)\right)^N}{1+e^{2\alpha}}
$$
I think I have found a sign error. On the left hand side we have
\begin{align*}
\sum_{m=0}^{N-1} \dfrac{ 2(N-1)! }{ (N-1-m)! m! } e^{ -2m \alpha + (N-1) \alpha } &= 2 e^{ (N-1) \alpha } \sum_{m=0}^{N-1} \binom{N-1}{m} \left( e^{ -2 \alpha } \right)^m \\ & = 2(e^{\alpha})^{N-1} \left( 1+e^{-2\alpha} \right)^{N-1}\\
&= 2 \left( e^\alpha + e^{-\alpha}\right)^{N-1} \\
&= 2 e^{-(N-1)\alpha}e^{+(N-1)\alpha} \left( e^\alpha + e^{-\alpha}\right)^{N-1}\\
&= 2 e^{-(N-1)\alpha}\left( e^{2\alpha} + 1 \right)^{N-1}
\end{align*}
Ok, no sign error. One just has to be very cautious.
 
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  • #5
Actually, there was a sign error somewhere when I calculated these sums with Mathematica to get an equation for the heat capacity of an infinite spin chain... It shouldn't be left in the equation above, but it's not impossible either.
 

Related to Summation formula from statistical mechanics

What is the summation formula from statistical mechanics?

The summation formula from statistical mechanics is a mathematical expression used to calculate the total energy of a system by summing up the contributions from all the individual particles or components of the system.

Why is the summation formula important in statistical mechanics?

The summation formula is important in statistical mechanics because it allows us to calculate the macroscopic properties of a system, such as energy and temperature, based on the microscopic behavior of its individual components.

What are the components of the summation formula?

The components of the summation formula include the energy of each individual particle or component, the number of particles or components in the system, and a weighting factor, such as the Boltzmann factor, that takes into account the probability of each energy state.

How is the summation formula derived?

The summation formula is derived from the principles of statistical mechanics, which use statistical methods to describe the behavior of a large number of particles. It is based on the assumption that the total energy of a system is equal to the sum of the energies of its individual components.

Can the summation formula be applied to any system?

Yes, the summation formula can be applied to any system, as long as its components can be described by a discrete set of energy states. It is commonly used in the study of gases, liquids, and solids, but can also be applied to more complex systems such as biological molecules or even the entire universe.

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