Hi everyone. I was pondering how big the concentration fluctuations of ATP at equilibrium would be in a (dead) cell, so started trying to apply some statistical mechanics. The results are
attached... I am interested in your impressions/thoughts/simplifications/ridicule, etc.
What is the system to which you are applying the grand canonical distribution? You specify the reservoir as "all other modes", but this is true for every mode in the sum. This means that the system is not clear - every term in the sum is resetting the definition of system and bath. This is going to cause problems.
ReplyDeleteThe 'system' (one mode) and 'bath' (all other modes) are in diffusive equilibrium, so they must have the same chemical potential. Each mode is treated separately, with the constraint that the total number of particles in every mode must sum to a constant number. The energy of a single particle in each mode depends only on the geometry of the box and the corresponding quantum numbers, so is independent of the occupancy of that mode (since any interactions are presumed to be very weak).
ReplyDeleteThe entirety of the box is a 'system' in contact with a thermal reservoir 'bath', but is unable to exchange particles with it (even if I introduce a small hole and allow the box to be treated with the grand canonical distribution the answer should come out the same, but now N is the thermal average number of particles in each box).
This is essentially how Schroder and Kittel treat the ideal gas, as best as I can tell.
I perceive that you are very keen to use tools that you have learned, but I also perceive that you are making your life hard by using them where they are not needed. I just don't think you need to invoke the chemical potential of orbitals at all here. If the state is an ideal gas state at some temperature for a box with set particle number, then you know how to write the state. The full state is a sum over all eigenmodes yes, but there is no need to consider fluctuations in individual mode populations because what you are trying to determine is the fluctuation at a point in space, summed over ALL modes. You end up calculating the fluctuations in a given mode, but you don't need this quantity, so why follow this path? All you need is the total spatial wavefunction in order to calculate its dispersion.
ReplyDeleteMind you I'm not saying you couldn't get there this way, just that it isn't the simplest way.
ReplyDeleteIt shall have to wait, but I'll try both ways!
ReplyDelete