I couldn't understand the concept of free energy landscape. Usually free energy is defined in the following way. $$F=-k_{B}T\ln Z=-k_{B}T\ln\sum e^{-\beta \epsilon}$$ where $\epsilon=\sum\frac{p_{i}^{2}}{2m}+V(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3},...,\vec{x}_{n})$. Here the summation over all the $\{\vec{p}_{i}\}$ and $\{\vec{x}_{i}\}$. So $F$ seems like a function of $\beta$ {so, $F=F(\beta)\}$. So where does the concept of landscape come here? The derivation of probability distribution comes in the following way$$P(\epsilon) \propto \Omega_{\text{system}}(\epsilon)\Omega_{\text{reservoir}}(E-\epsilon)$$ where $E=E_{\text{reservoir}}+\epsilon$, with $E$ total energy of system+reservoir. Expressing in terms of entropy we get $$P(\epsilon)\propto\exp\bigg(\frac{S_{\text{system}}(\epsilon)}{k_{B}}\bigg)\exp\bigg(\frac{S_{\text{reservoir}}(E-\epsilon)}{k_{B}}\bigg)$$ Taylor expansion of $S_{\text{reservoir}}(E-\epsilon)=S_{\text{reservoir}}(E) - \epsilon\bigg(\frac{\partial S_{\text{reservoir}}}{\partial E}\bigg) +O(\epsilon^{2})\sim S_{\text{reservoir}}(E)-\frac{\epsilon}{T}$ \begin{align} P(\epsilon) &\propto \exp\bigg(\frac{S_{\text{system}}(\epsilon)}{k_{B}}\bigg)\exp\bigg(\frac{S_{\text{reservoir}}(E)-\frac{\epsilon}{T}}{k_{B}}\bigg) \\ &\propto\exp\bigg(\frac{S_{\text{system}}(\epsilon)-\frac{\epsilon}{T}}{k_{B}}\bigg)\exp\bigg(\frac{S_{\text{reservoir}}(E)}{k_{B}}\bigg)\\ &\propto\exp\bigg(-\frac{F}{k_{B}T}\bigg) \end{align} So, \begin{align}F&\propto-k_{B}T\ln P(\epsilon)\\ &\propto-k_{B}T\ln P(\epsilon(\vec{p}_{1},\cdots,\vec{p}_{n};\vec{x}_{1},\cdots,\vec{x}_{n} ))\\ &=-k_{B}T\ln P(\vec{p}_{1},\cdots,\vec{p}_{n};\vec{x}_{1},\cdots,\vec{x}_{n} ). \end{align} I ended up this result because of degeneracy of the system in the given energy $\epsilon$ If I assume, there is no degeneracy of the system then, $P(\epsilon)\propto\Omega_{\text{reservoir}}(E-\epsilon)\propto \exp(-\beta\epsilon)$ . Free energy becomes functions of all the momenta and position coordinates only when there is a degeneracy in the given energy of the system. I have two questions,
- Is free energy landscape a function of momenta and position coordinates? If so, is my derivation correct?
- Again the dependence of position and momenta coordinates comes into play in the free energy expression only when there is a degeneracy in the energy level of the system. Can I say free energy surface makes sense only when there is a degeneracy in system's energy levels?
I am asking in the context polymer/protein folding problem where one encounter free energy landscape.