GATE Physics Thermodynamics and Statistical Physics Syllabus

Thermodynamics and Statistical Physics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Thermodynamics and Statistical Physics in GATE Physics, not a summary of it.

1. 1

   Laws of Thermodynamics

   2 topics

   • Macrostates and Microstates
   • Phase Space
2. 2

   Ensembles

   3 topics

   • Partition Function
   • Free Energy
   • Calculation of Thermodynamic Quantities
3. 3

   Classical and Quantum Statistics

   3 topics

   • Degenerate Fermi Gas
   • Black Body Radiation and Planck's Distribution Law
   • Bose-Einstein Condensation
4. 4

   First and Second Order Phase Transitions

   2 topics

   • Phase Equilibria
   • Critical Point

Thermodynamics and Statistical Physics flashcards for GATE Physics

24 of 51 cards from the Thermodynamics and Statistical Physics deck — real questions with worked answers.

1. What is a microstate in statistical mechanics?

   A microstate is a specific detailed configuration of a system in which the exact quantum state (position and momentum, or quantum numbers) of every particle is fully specified.

2. What is a macrostate, and how does it differ from a microstate?

   A macrostate is the description of a system in terms of macroscopic variables (e.g. $N$, $V$, $E$, $T$, $P$). A single macrostate generally corresponds to a very large number of microstates that all share the same macroscopic properties.

3. State Boltzmann's entropy formula relating entropy to the number of microstates.

   $$S = k_{B} \ln \Omega$$ where $\Omega$ is the number of accessible microstates and $k_{B}$ is Boltzmann's constant.

4. What is the fundamental postulate of statistical mechanics for an isolated system?

   For an isolated system in equilibrium, all accessible microstates of a given energy are equally probable (the postulate of equal a priori probabilities).

5. For $N$ distinguishable two-state particles, how many microstates correspond to the macrostate with $n$ particles in the excited state?

   $$\Omega(n) = \binom{N}{n} = \frac{N!}{n!\,(N-n)!}$$

6. What is phase space for a system of $N$ particles in three dimensions?

   Phase space is the $6N$-dimensional space spanned by the $3N$ position coordinates and $3N$ momentum coordinates $(q_{1},\dots,q_{3N},p_{1},\dots,p_{3N})$; a single point represents the complete microstate of the system.

7. What volume of phase space does one quantum microstate occupy, and why?

   Each microstate occupies a volume $h^{3N}$ in phase space (i.e. $h$ per degree of freedom), as a consequence of the uncertainty principle $\Delta q\,\Delta p \sim h$.

8. State Liouville's theorem regarding phase space density.

   The phase-space density $\rho$ of an ensemble of systems behaves like an incompressible fluid: $\frac{d\rho}{dt} = 0$, i.e. the local density along a trajectory is constant in time.

9. Define the single-particle partition function $Z_{1}$ in the canonical ensemble.

   $$Z_{1} = \sum_{i} e^{-\beta \varepsilon_{i}}$$ where the sum runs over single-particle states $i$ with energy $\varepsilon_{i}$ and $\beta = \frac{1}{k_{B}T}$.

10. How is the $N$-particle partition function related to $Z_{1}$ for an ideal gas of indistinguishable particles (classical limit)?

    $$Z_{N} = \frac{Z_{1}^{N}}{N!}$$ The $N!$ corrects for the indistinguishability of identical particles (Gibbs factor).

11. What is the probability of finding a system in a microstate of energy $E_{i}$ in the canonical ensemble?

    $$P_{i} = \frac{e^{-\beta E_{i}}}{Z}, \qquad Z = \sum_{i} e^{-\beta E_{i}}$$ This is the Boltzmann (canonical) distribution.

12. How is the average internal energy obtained from the partition function $Z$?

    $$\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} = k_{B}T^{2}\frac{\partial \ln Z}{\partial T}$$

13. Write the thermal de Broglie wavelength $\lambda$ used in the partition function of an ideal gas.

    $$\lambda = \frac{h}{\sqrt{2\pi m k_{B} T}}$$

14. Give the single-particle translational partition function for an ideal gas in a volume $V$.

    $$Z_{1} = \frac{V}{\lambda^{3}} = V\left(\frac{2\pi m k_{B} T}{h^{2}}\right)^{3/2}$$

15. Define the Helmholtz free energy and give its relation to the partition function.

    $$F = U - TS, \qquad F = -k_{B}T \ln Z$$

16. Define the Gibbs free energy $G$ in terms of $U$, $T$, $S$, $P$, $V$.

    $$G = U - TS + PV = H - TS = F + PV$$

17. From the Helmholtz free energy $F$, how do you obtain entropy, pressure, and internal energy?

    $$S = -\left(\frac{\partial F}{\partial T}\right)_{V}, \quad P = -\left(\frac{\partial F}{\partial V}\right)_{T}, \quad U = F + TS$$

18. What is the grand potential $\Phi$ and its relation to the grand partition function $\mathcal{Z}$?

    $$\Phi = -k_{B}T \ln \mathcal{Z} = -PV, \qquad \mathcal{Z} = \sum_{N,i} e^{-\beta(E_{i}-\mu N)}$$

19. State the Sackur–Tetrode equation for the entropy of a monatomic ideal gas.

    $$S = N k_{B}\left[\ln\!\left(\frac{V}{N}\left(\frac{2\pi m k_{B}T}{h^{2}}\right)^{3/2}\right) + \frac{5}{2}\right]$$

20. What are the average energy and heat capacity of a classical monatomic ideal gas (equipartition)?

    $$U = \frac{3}{2}N k_{B}T, \qquad C_{V} = \frac{3}{2}N k_{B}$$

21. State the equipartition theorem.

    In classical equilibrium, each quadratic degree of freedom in the Hamiltonian contributes an average energy of $\frac{1}{2}k_{B}T$ to the internal energy.

22. Write the Maxwell–Boltzmann speed distribution $f(v)$.

    $$f(v) = 4\pi n \left(\frac{m}{2\pi k_{B}T}\right)^{3/2} v^{2}\, e^{-\tfrac{mv^{2}}{2k_{B}T}}$$

23. Give the most probable speed, mean speed, and rms speed of a Maxwell–Boltzmann gas.

    $$v_{p}=\sqrt{\frac{2k_{B}T}{m}},\quad \langle v\rangle=\sqrt{\frac{8k_{B}T}{\pi m}},\quad v_{rms}=\sqrt{\frac{3k_{B}T}{m}}$$

24. Write the Fermi–Dirac distribution function.

    $$\langle n_{i}\rangle = \frac{1}{e^{\beta(\varepsilon_{i}-\mu)}+1}$$

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