Fluctuation Dissipation Theorem

In summary: However, as it stands, the given information is not sufficient to determine the value of qp. In summary, the given system of coupled Langevin equations represents a model for protein synthesis, with r representing the concentration of mRNA and p representing the concentration of protein. The condition for Poisson statistics only applies to r and cannot be used to determine the value of qp. More information or simplifying assumptions would be needed to solve for qp.
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Homework Statement


We have a system of two coupled Langevin equations
dr/dt=kr-yrr+nr(t)
dp/dt=kpr-ypp+np(t)
where the ki,yi are constants and ni(t) are noise terms satisfying <ni(t)>=0 and <ni(t')ni(t'')>=qiδ(t'-t'') (this is zero if the two indices differ).

The physical background of these is that it is a model of protein synthesis - r represents the concentration of mRNA and p the concentration of a protein.

Homework Equations


The concentration of mRNA should obey Poisson statistics in the steady state, which means that in the long time limit we have <r>=<r2>-<r>2.

The Attempt at a Solution


I can solve the first equation for <r> and <r2> and apply the above condition to obtain qr=2kr. I can also then plug my expression for r(t) into the second to obtain expressions for <p> and <p2>, however p does not obey Poisson statistics at long times My issue is in finding the value of qp - the answer should be qp=2kpkr/yr but I cannot understand how to extract this from the condition that only r obeys Poisson statistics at long times.
 
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  • #2


Based on the given equations, it seems that the noise term for p, np(t), is not included in the condition for Poisson statistics. This means that the fluctuations in p are not necessarily described by a Poisson distribution. Therefore, it is not possible to determine the value of qp from the given information.

To understand this further, we can look at the equations for <p> and <p2>:

<p> = kp<r> - yp<p> + <np(t)>

<p2> = kp<r2> - yp<p2> + <np(t)2>

Since <p> and <p2> both contain the noise term <np(t)>, they cannot be directly related to the condition for Poisson statistics. The condition only applies to <r> and <r2>, which do not contain the noise term for p.

In order to determine the value of qp, we would need more information about the system, such as the specific values of ki and yi. Alternatively, we could make some assumptions or approximations to simplify the equations and make it possible to solve for qp.
 

1. What is the Fluctuation Dissipation Theorem?

The Fluctuation Dissipation Theorem (FDT) is a fundamental principle in statistical mechanics that describes the relationship between the fluctuations and dissipation of a system in thermal equilibrium. It states that the ratio of the fluctuation in a system's thermodynamic variable to the dissipation of that same variable is equal to the temperature of the system.

2. How is the Fluctuation Dissipation Theorem derived?

The FDT is derived mathematically using the principles of equilibrium thermodynamics and the fluctuation-dissipation theorem in linear response theory. It can also be derived from the second law of thermodynamics, which states that the total entropy of a system and its surroundings cannot decrease.

3. What are some applications of the Fluctuation Dissipation Theorem?

The FDT has numerous applications in various fields of science, including physics, chemistry, biology, and engineering. It is commonly used in the study of phase transitions, critical phenomena, and the behavior of complex systems such as glasses and polymers. It has also been applied in the development of technologies such as magnetic resonance imaging and molecular motors.

4. What are the limitations of the Fluctuation Dissipation Theorem?

The FDT assumes that the system is in thermal equilibrium, which may not always be the case in real-world scenarios. It also assumes linearity and time-reversal symmetry, which may not hold for systems with strong non-equilibrium effects or in the presence of external forces. Additionally, the FDT may not be applicable to systems with long-range interactions or in the absence of thermal fluctuations.

5. How does the Fluctuation Dissipation Theorem relate to other thermodynamic principles?

The FDT is closely related to other thermodynamic principles, such as the second law of thermodynamics, which governs the direction of heat flow and the irreversibility of processes. It is also related to the fluctuation theorem, which describes the probability of observing fluctuations in a system. The FDT provides a useful tool for understanding the behavior of systems in thermal equilibrium and can help reconcile macroscopic thermodynamic principles with microscopic fluctuations.

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