Fastest Frozen Temperature for a Thermodynamic System

For a thermodynamic system obeying both the equipartition theorem in high temperature and the third law in low temperature, the curve showing relationship between the specific heat and the temperature has two common behaviors:\ it terminates at zero when the temperature is zero Kelvin and converges to a constant as temperature is higher and higher. Since it is always possible to find the characteristic temperature $T_{C}$ to mark the excited temperature as the specific heat almost reaches the equipartition value, it is reasonable to find a temperature in low temperature interval, complementary to $T_{C}$. The present study reports a possibly universal existence of the such a temperature $\vartheta$, defined by that at which the specific heat falls \textit{fastest} along with decrease of the temperature. For the Debye model of solids, above the temperature $\vartheta$ the Debye's law starts to fail.


I. INTRODUCTION
In classical statistical mechanics, we have the equipartition theorem that applies for a system of many particles that obey classical mechanics. Suppose the energy = + + + ... of a molecule is expressed as a sum of independent terms , , , ... each referring to a different degree of freedom, and all these terms of the energy are expressed as squared terms of type α j ξ 2 j with ξ j being the j-th generalize position or momentum and α j being the j-th coefficient independent of all ξ j (j = 1, 2, 3, ..., f ), we can prove that average energy per molecule u ≡ = f k B T /2 where k B is Boltzmann's constant and T is the temperature in Kelvins. This embodies the equipartition theorem that formally states that when a large number of indistinguishable, quasi-independent particles whose energy is expressed as the sum of f squared terms come to equilibrium, the average internal energy per particle is f times k B T /2. [1][2][3][4][5] For instance, a monatomic ideal gas has only three translational degrees of freedom, then u = 3k B T /2. The diatomic gases are of three translational degrees of freedom and two rotational ones and two vibrational ones, and we should have u = 7k B T /2; but at room temperature, only translational and rotational degrees of freedom are activated so u = 5k B T /2 is agreement with experiments. The vibrational degrees of freedom within a diatomic molecule are frozen out at room temperature. Generally speaking, the equipartition theorem is not always valid; it applies for a degrees of freedom that can be freely excited. At a given temperature T , there may be certain degrees of freedom which are more or less frozen due to quantum mechanical effects. [1][2][3][4][5] In order to capture the essential cause to possible violation of the equipartition theorem, we can construct a model system consisting of indistinguishable, quasi-independent particles each of them has continuous energy level except the ground state one, then explore its behaviors of the heat capacity. The key finding is that, when temperature rises from zero Kelvin, there is a definite temperature at which the heat capacity increases rapidly, and vice versa. In other words, once the temperature is lower than this one, the relevant degree of freedom is almost frozen; and in this sense we can take this fastest frozen temperature as the frozen temperature itself as well. In present paper, we confine ourself to the case the non-degenerate gases such that the Boltzmann statistics can apply.
In textbooks, for a specific degree of freedom the frozen criterion is qualitatively expressed as T T C where T C is the characteristic temperature defined via in which ε 0 and ε 1 are, respectively, the ground state and first excited energy of the degree of freedom under study. This characteristic temperature T C is the temperature at which the degree of freedom is almost activated and it makes a significant contribution toward the specific heat of the system. For instance, for the vibrational degrees of freedom, the value of the heat capacity at T C is "about 93 per cent of the equipartition value". [2] For simplicity, in rest part of the present paper, we will use specific heat defined by arXiv:2001.03007v1 [cond-mat.stat-mech] 9 Jan 2020 In our approach, all other parameters remain unchanged except the temperature T , the partial derivative "∂" above can be replaced by the derivative "d". The third law of thermodynamics implies that all degrees of freedom are completely frozen at zero Kelvin [1][2][3][4][5] lim T →0 We will show that for a given degree of freedom, the existence of the characteristic temperature T C (1) accompanies a fastest frozen temperature ϑ, which is defined by the maximum of the dc/dt dc dT The paper is organized in the following. In section II, we present a theorem based on a model system to explicitly demonstrate the existence of the fastest frozen temperature ϑ. In section III, some examples are given. The final section IV concludes this study.

II. EXISTENCE OF FASTEST FROZEN TEMPERATURE ϑ: A THEOREM
Construct a model system that has a kind of degrees of freedom whose energy levels are simply continuous except the ground state one which is isolated from the rest, i.e., in which the spacing between 1 , 2 , 3 , ...is negligible but 1 is appreciably different from ground state one 0 which can be conveniently chosen to to be zero, 0 = 0, and the density of states can be A (D−2)/2 , in which D = 1, 2, 3, 4, 5, 6 can be understood as the number of the dimension of the spaces, where coefficient A can be set to be unity. Utilize the Boltzmann statistics, the partition function is with β = (k B T ) where Γ(s, x) = ∞ x t s−1 e −t dt is the incomplete gamma function and Γ(s) = Γ(s, 0) the ordinary gamma function. This problem is analytically tractable, but the relevant expressions are lengthy. The energy per particle and the specific heat c are, respectively, determined by We do not explicitly show their expressions. We compute c(T ) and dc(T )/dT , respectively, for D = (1, 2, 3, 4, 5, 6), and find they are similar. Thus, we plot c(T ) and dc(T )/dT for D = 3 in Fig.1 only. It is clearly that the c( The values of the fastest frozen temperature ϑ and ratios of two specific heats c(ϑ)/c(T C ) are listed in following Table.  Since c(ϑ)/c(T C ) are all significantly smaller than 1, we can therefore take ϑ as a quantitative criterion for the frozen temperature.
The results above can be summarized in a theorem: For a degreee of freedom whose energy levels are continuous except for the ground state one, it has a fastest frozen temperature in the course the degreee of freedom is freezing.

III. FASTEST FROZEN TEMPERATURE ϑ: EXAMPLES
In this section, we present some examples to demonstrate the theorem above. Example 1: ϑ of vibrational degrees of freedom for a diatomic gas. The energy level is n = (n + 1/2) ω, (n = 0, 1, 2, ...), where is the Planck's constant and ω is the vibrational frequency. The partition function is, [2] With the characteristic temperature T C = ω/k B , we have the mean vibrational energy u and specific heat c, respectively, The specific heat at T C is c(T C ) = 0.921k B lim T →∞ c(T ) = k B . The fastest frozen temperature ϑ = 0.223T C at which c(ϑ) = 0.231k B from which we see that vibrational degrees of freedom are thermally depressed. The value T C of different diatomic gases is of order 10 3 K, [2] and we have then ϑ ∼ (200 − 300) K so the vibrational degrees of freedom at room temperature is almost frozen. We plot the specific heat c(T ) and dc/dt against temperature in Fig.2.
Example 2: ϑ of rotational degrees of freedom for a heteronuclear diatomic gas. The energy level is l = l(l + 1) 2 /2I, (l = 0, 1, 2, ...), where I is the moment of inertia. The partition function is, Note that in textbooks [1][2][3][4][5] the characteristic temperature T C = 2 / (2Ik B ) rather than (1). We follow this convention. The mean vibrational energy u and specific heat c are, respectively, determined by where The numerical calculations give both c(T C ) = 1.07k B > lim T →∞ c(T ) = k B and the fastest frozen temperature ϑ = 0.390T C at which c(ϑ) = 0.452k B . Once T decreases from ϑ the rotational degrees of freedom is rapidly frozen out. The value of T C , for example, for HCl is about 15 K, [2] we have ϑ = 12 K so the rotational degrees of freedom at room temperature is freely excited. We plot the specific heat c(T ) and dc/dt against temperature in Fig.3.
The mean vibrational energy u and specific heat c are, respectively, determined by We still follow the convention of taking the rotational characteristic temperature T C = 2 / (2Ik B ) as done in standard textbooks. [1][2][3][4][5] The numerical calculations give both the fastest frozen temperature ϑ = 1.33T C at which c(ϑ) = 0.305k B and c(T C ) = 0.117k B << k B = lim T →∞ c(T ). From these results, we see that this T C is right the frozen temperature rather than the excited one. The specific heat c(T ) and dc/dt against temperature in Fig.4.
The value of T C for H 2 is 85 K and the excited temperature is much higher. [2] Note that at T = 0 K, hydrogen contains mainly para-hydrogen which is more stable, and in general the concentration ratio of the ortho-to parahydrogen in thermal equilibrium is given by [2] from which we obtain r(T C ) = 7.44, r(ϑ) = 4.74, r(300) = 3.01.
These numbers indicate again that both ϑ and T C are actually frozen temperature rather than the activated ones, while the room temperature T ≈ 300 K= 3.53T C is the activated temperature at which the ratio r ≈ 3, and thus "the name ortho-is given to that component which carries the larger statistical weight". [2] In fact, our definition of the characteristic temperature (1) is 3T C = 2 − 0 and 5T C = 3 − 1 , respectively, for ortho-and para-hydrogen. Other results are shown in Fig.4. We leave some exercises to the readers to determine the fastest frozen temperatures for, e.g., two-level system, gas of deuterium molecule and model system (5)-(6) with fractal dimensions D. The results are similar.

IV. CONCLUSION
The known frozen temperature for a degree of freedom is qualitatively expressed as T T C which is usually the characteristic temperature defined via k B T C ≡ ε 1 − ε 0 in which ε 0 and ε 1 are, respectively, the ground state and first excited energy of the given degree of freedom. This characteristic temperature T C is actually the excited temperature for the degree of freedom because the heat capacity is at least 90% of the equipartition value. A welldefined temperature at which the specific heat falls fastest along with decrease of the temperature is identified, which can be taken as the quantitative criterion of the frozen temperature itself. It is specially useful for some systems such as a mixture of ortho-and para-hydrogen molecules, in which the characteristic temperature is hard to define.    Curves for specific heat (solid) and its derivative with respect to temperature (dashed) for the rotational degrees of freedom of 1 : 3 mixture of para-hydrogen and ortho-hydrogen molecules, and the auxiliary dotted lines and arrows are guide for the eye. The equipartition value of the specific heat is 1kB, and c(TC ) = 0.117kB is much smaller than 1kB. The dc/dT has a global maximum at ϑ = 1.33TC at which the specific heat c(ϑ) = 0.305kB. The conventional definition of characteristic temperature TC = 2 /(2IkB) is nothing but a unit, and once we use our definition (1), we have for ortho-and para-hydrogen, respectively, c(ϑ ortho ) = 0.907kB and c(ϑpara) = 0.999kB.