Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 37A (Statistical Physics)

The extension of the one-dimensional freely jointed chain isAs the integer $n_{+}$ runs from $0$ to $n$, the possible extensions are thereforeFor fixed $n_{+}$ and $n_{-}=n-n_{+}$, choosing which $n_{+}$ of the $n$ links point in the positive direction determines the microstate. The state degeneracy is consequently the binomial coefficient

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The mean extension in an ensemble of $A$ molecules iswith constraints ${\sum }_i{a}_i=A$ and ${\sum }_i{a}_i{l}_i=AL$. For occupancies $\{a_i\}$, the number of assignments of the $A$ distinguishable ensemble members to the states is the multinomial coefficientThus maximizing the probability at fixed $A$ and $L$ is equivalent to maximizing the stated Lagrange multiplier expression.

Because $A\gg 1$, the Stirling formula gives $\log (a_i!)\simeq a_i\log a_i-a_i$. Differentiating with respect to each $a_i$ givesNormalization by ${\sum }_i{a}_i=A$ therefore yields the probability mass functionHere $Z$ is the fixed-tension partition function that normalizes the probabilities.

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Several microstates can have the same extension. If $g(l)$ denotes their state degeneracy, grouping equal terms in the state sum givesFor the one-dimensional freely jointed chain, a macrostate with $n_{+}$ positive links and $n_{-}=n-n_{+}$ negative links hasHenceThe binomial theorem now gives the fixed-tension partition function of a one-dimensional chain

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Substituting the fixed-tension partition function of a one-dimensional chain into the given free-energy definition givesDifferentiation with respect to the parameter conjugate to extension givesThe same result follows directly from the expected value under $p_i=e^{\tau l_i}/Z$:Consequently

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If a physical tension $f$ acts on a state of extension $l_i$, its mechanical energy is $-f{l}_i$. The canonical ensemble therefore assigns that state the Boltzmann factorComparison with the statistical weight $e^{\tau l_i}$ givesEquivalently, $f$ is the thermodynamic conjugate variable to $L$ in the Gibbs free energy, just as pressure is conjugate to volume.

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The condition $L\ll na$ implies $|\tau a|\ll 1$. The small-argument expansion of the hyperbolic tangent gives $\tanh x=x+O(x^3)$, soUsing $f=k_{B}T\tau$ yields the entropic Hooke law for a one-dimensional chainThus the effective spring constant is $k_{B}T/(n{a}^2)$.

At fixed tension, the exact extension isIncreasing the temperature decreases the positive argument of the hyperbolic tangent, so the chain contracts. In the small-extension regime this becomeswhich is inversely proportional to $T$.

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