Why do systems with greater energy fluctuation have better heat dissipation ability?

Question

In canonical ensemble ($NVT$), it can be shown that the fluctuations in energy is related to the specific heat of the system according to the formula,

$$ \left( \Delta E \right)^2 = k_B T^2 C_V $$ where $T$ is the temperature of the system and $C_V$ is the specific heat of the system at constant volume. This is referred to as the fluctuation-dissipation theorem which comes in various manifestations.

Interpretation

If I have two systems (1 and 2) such that $C_{V_1} > C_{V_2}$, then adding the same amount of heat to both systems, we can expect a greater rise in temperature in system 2 (system with lower specific heat). Therefore, $\left(\Delta T\right)_1 < \left(\Delta T\right)_2$.

According to the fluctuation dissipation theorem above, system with higher $C_V$ can have larger energy fluctuations. The higher fluctuations in energy typically implies that the system can access greater region of phase-space (state-space), therefore it has a better ability to redistribute the energy to all the particles in the system and therefore, the average increase in energy per particle is less, and therefore temperature rise will also be less compared to a system with smaller specific heat.

Is this a valid interpretation of the fluctuation-dissipation theorem?

In principle, I want to understand why does larger fluctuations in equilibrium systems contribute to better dissipation within them.

Is it probably linked to the accessibility of the system to more configurations and hence leading greater dissipation?

Answer 1

Your interpretation is on the right track. Larger fluctuations contribute to better dissipation for several reasons:

Why do larger fluctuations in equilibrium systems contribute to better dissipation within them?

• As you suggested, larger fluctuations allow the system to access a wider range of microstates. This increased accessibility means the system has more ways to redistribute energy internally.
• Systems with larger fluctuations typically have more active degrees of freedom or stronger coupling between them. This allows energy to be more easily transferred between different modes or particles within the system.
• The fluctuation-dissipation theorem relates the system's response to small perturbations to its equilibrium fluctuations. Larger fluctuations indicate a greater capacity to absorb and dissipate small disturbances.

Is it probably linked to the accessibility of the system to more configurations and hence leading greater dissipation?

Yes, this is a valid way to think about it.

• More accessible configurations mean the system has more ways to distribute energy among its constituents.
• This greater number of available states allows the system to more easily find configurations that accommodate new energy inputs without significantly changing its macroscopic properties (like temperature).
• The ability to redistribute energy more effectively across many configurations is essentially what we mean by dissipation at the microscopic level.

More:

The entropy $S$ of a system is related to the density of states through the Boltzmann formula ($S = k_B \ln(\Omega$)) where $\Omega$ is the number of microstates available to the system, which is directly related to the integrated density of states over the accessible energy range.

The specific heat $C_V$ can be derived from the entropy:

\begin{equation} C_V = T\left(\frac{\partial S}{\partial T}\right)_V \end{equation}

This shows that systems with a higher density of states (and thus higher entropy) will generally have a higher specific heat.

• Higher density of states $\rightarrow$ more accessible microstates $\rightarrow$ larger fluctuations

• Larger fluctuations $\rightarrow$ better ability to redistribute energy $\rightarrow$ more effective dissipation

• Higher specific heat $\rightarrow$ larger energy fluctuations (as per the fluctuation-dissipation theorem)