The Grüneisen parameter [Wiki], γ_th, is defined as a change in P by a change in density of E at constant V. Namely,

To put it plainly, γ_th is the ratio of thermal pressure to thermal energy [Wiki]. Or, it describes how much P increases when the matter is heated.

It is expressed by:

If  γ_th is defined by this equation, it is called the thermodynamic Grüneisen parameter.

To put it plainly, what Eq (2) means is as follows.

• If E increases, the matter wants to increase its V due to α, because T should increase with E increase. Therefore, γ_th is proportional to α.
• If K_T is higher, the potential V increase that the E increase wants to causes larger P increase. Accordingly, γ_th is proportional to K_T.
• If the C_V is larger, T increase becomes smaller for a given E increase, and therefore, γ_th is inversely proportional to C_V.

In the case of MgSiO₃ bridgmanite at ambient conditions, α = 2.0×10^-5 K^-1, K_T = 260×10⁹ Pa, V = 24×10^-6 m³ mol^-1, C_V is 80 J K^-1 mol^-1, and therefore, γ_th = 1.1.

Note that γ_th is a dimensionless parameter.

Eq. (2) is obtained from Eq. (1) by the following way.

The numerator of Eq (3) is:

The denominator of Eq (3) is:

Using Eqs (3)-(5), Eq. (2) is obtained from Eq. (1) as:

The definition of the Grüneisen parameter, Eq. (1), is the Mie-Grüneisen EOS [Wiki] itself:

There are other ways to define the Grüneisen parameter.
For example, the mode Grüneisen parameter, γ_i, is defined as:

where ω_i is the angular frequency of the lattice vibration of the mode i.
The Debye gamma, γ_D, is defined as:

where ω_D is the Debye cut-off (angular) frequency [Wiki], and Θ_D is the Debye temperature [Wiki].

We will discuss the mode Grüneisen parameter and Debye gamma elsewhere.