Energy Fluctuation in Canonical Ensemble

Show that for a canonical ensemble the fluctuation of energy in a system of constant volume is related to the specific heat and, hence, deduce that the specific heat at constant volume is nonnegative.

SOLUTION

First solution: For a canonical ensemble:

(1)

where β = 1/τ. On the other hand,

(2)

Differentiating (2), we obtain

(3)

By inspecting (1)–( 3), we find that

(4)

Now, the heat capacity at constant volume, C_v, is given by

(5)

Therefore, comparing (4) and (5), we deduce that, at constant volume,

(6)

or in standard units

(7)

Since

then

Second solution: A more general approach may be followed which is applicable to other problems. Because the probability ω of finding that the value of a certain quantity X deviates from its average value ⟨X⟩ is proportional to e^{S(X - ⟨X⟩)} and denoting x = X - ⟨X⟩, we can write

(8)

Note that ⟨X⟩ = 0. The entropy has a maximum at x = 0. Expanding S(x), we obtain

where

so λ > 0. The probability distribution

(9)

⟨x²⟩ = λ, so

(10)

If we have several variables,

(11)

If the fluctuations of two variables x_i, x_k are statistically independent,

The converse is also true: If ⟨x_ix_k⟩ = 0, the variables x_i and x_k are statistically independent. Now for a closed system we can write

(12)

where S₀ is the total entropy of the system and ∆S₀ is the entropy change due to the fluctuation. On the other hand,

(13)

where W_min is the minimum work to change reversibly the thermodynamic variables of a small part of a system (the rest of the system works as a heat bath), and τ is the average temperature of the system (and therefore the temperature of the heat bath). Hence,

However,

(14)

where ∆E, ∆S, and ∆V are changes of a small part of a system due to fluctuations and τ, P are the average temperature and pressure. So,

(15)

Expanding ∆E (for small fluctuations) gives

(16)

Substituting (16) into (14), we obtain

(17)

where we used

So, finally

(18)

Using V and τ as independent variables we have

(19)

Substituting (19) into (18), we see that the cross terms with ∆V ∆τ cancel (which means that the fluctuations of volume and temperature are statistically independent, ⟨∆V. ∆τ ⟩ = 0):

(20)

Comparing (20) with (10), we find that the fluctuations of volume and temperature are given by

(21)

To find the energy fluctuation, we can expand ∆E:

(22)

where we used ⟨∆V. ∆τ ⟩ = 0. Substituting  and from (21), we obtain a more general formula for

(23)

At constant volume (23) becomes

the same as before.

مواضيع ذات صلة

مصطلحات الفيزياء النووية

إزاحة حمراء REDSHIFT

المركبات النانوية (Nanocomposites)

مولد فان دي جراف الالكتروستاتي

التطويرات التقنية الحديثة في المعجلات الالكتروستاتية

معجلات الالكترونات الالكتروستاتية

معجلات الفان دي جراف الترادفية

الحد من زيادة الطاقة في معجل فان دي جراف

أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

مبدأ تشغيل معجل الفان دي جراف

القضايا الأخلاقية المتعلقة بتكنولوجيا النانو