The constant-volume specific heat, often denoted as $C_v$, is a fundamental thermodynamic property of a substance. $C_v$ represents the amount of heat energy required to raise the temperature of a unit mass (or mole) of a substance by one degree Celsius (or Kelvin) when the volume is held constant. In simpler terms, it measures how much heat a substance can absorb before its temperature significantly increases, under conditions where it cannot expand.

Here provides a clear explanation of how to calculate root mean square (RMS) fluctuations and how they relate to thermodynamic properties, particularly in the context of computer simulations.

Here's a breakdown of the key points:

1. Definition of RMS Deviation:

• The RMS deviation, denoted as σ(A), quantifies the spread of a quantity A around its mean.
• It's defined as the square root of the variance of A, which is the average of the squared deviation of A from its mean.
• The formula given is:
  • $\sigma^2( A )=\left\langle\delta A ^2\right\rangle_{ens}=\left\langle A ^2\right\rangle_{ens}-\langle A \rangle_{ens}^2$
  • where $\delta A = A -\langle A \rangle_{ens}$, and $ens$ represents the ensemble average.

2. Thermodynamic Properties and Fluctuations:

• The excerpt highlights the connection between RMS fluctuations and important thermodynamic properties:
  • Constant-volume specific heat capacity ($C_V$)
  • Constant-pressure specific heat capacity ($C_P$)
  • Thermal expansion coefficient ($α_P$)
  • Isothermal compressibility ($β_T$)
  • Thermal pressure coefficient ($γ_V$)
• It is pointed out that the relationship $\alpha_P=\beta_T \gamma_V$ means that only two of the last three quantities are independant.
• It also highlights the difference between instantaneous mechanical quantities and thermodynamic ensemble averages.

3. Canonical Ensemble and Specific Heat:

• The canonical ensemble (NVT) is emphasized as a suitable framework for computing fluctuations.
• A key result is the relationship between the variance of the Hamiltonian (H) and the constant-volume specific heat:
  • $\left\langle\delta H ^2\right\rangle_{N V T}=k_{B} T^2 C_V$
  • This equation allows for the calculation of $C_V$ from the fluctuations of the Hamiltonian in a simulation.
• It is important to remember that in this context, the energy variable "E" from general thermodynamic equations, is represented by the Hamiltonian "H" in the context of computer simulations.

4. Caution with Instantaneous Pressure:

• The excerpt warns against directly applying fluctuation formulas to instantaneous pressure (P) in the same way as for the Hamiltonian.
• This is because the instantaneous pressure in a simulation is not directly equivalent to the thermodynamic pressure, which is an ensemble average.
• Therefore, the formula $\sigma^2(P)=\left\langle\delta P^2\right\rangle=k_{ B } T / V \beta_T$ is not as straightforward to use as the specific heat formula.

🧠Simulates a system in the canonical ensemble (NVT) and calculates the specific heat