PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Statistical mechanics and thermodynamics → Classical thermodynamics ↓


Entropy and free energy


Understanding the concepts of entropy and free energy is important in the study of thermodynamics and statistical mechanics. These concepts provide information about the nature of energy transformations and the direction of time in the physical processes of the universe. Let's take a deeper look at these topics and explore them through examples, both visually and in simple numerical ways.

Entropy: a measure of disorder

Entropy is often defined as a measure of the disorder or randomness in a system. It is a central concept in the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time. In simple terms, systems tend to evolve toward a state of greater disorder or randomness.

Understanding entropy from a simple example

Consider a box that is divided into two compartments by a removable wall. One side contains gas molecules, while the other side is empty. Initially, the gas molecules are on one side of the box. Once the wall is removed, the gas molecules will spread out to fill both compartments. When the gas molecules distribute themselves more evenly throughout the box, entropy increases. The initial state was more ordered (low entropy), and the final state was more disordered (high entropy).

Initial state: |gas|empty|
Final state: |gas|gas|

Temperature and entropy changes

The change in entropy can also be related to the heat absorbed or released by a system and its temperature. This relationship is given as:

ΔS = Q/T

where ΔS is the change in entropy, Q is the heat exchanged, and T is the temperature in Kelvin.

Entropy of mixing

When two different gases are allowed to mix, the entropy of the system increases. This is because there are now more ways to arrange the molecules of the combined gases. Let us consider two gases, A and B, initially separated by a partition.

When the partition is removed, the molecules of gas A and gas B mix together. This mixing process increases entropy because the number of possible microscopic states of the system has increased. Each molecule has more locations, making the system as a whole more disordered.

Initial: |a|b|
Mixed: | ABABAB |

Free energy: Gibbs and Helmholtz

While entropy describes the disorder in a system, free energy provides a useful measure for predicting the direction of spontaneous change. In thermodynamics, there are two commonly used types of free energy: Gibbs free energy and Helmholtz free energy.

Gibbs free energy

The Gibbs free energy, G, is defined as:

G = H – TS

where H is enthalpy, T is temperature, and S is entropy. Gibbs free energy is particularly useful in predicting the direction of chemical reactions at constant pressure and temperature.

If the change in Gibbs free energy is negative then the reaction is spontaneous:

ΔG < 0

Consider a simple example of ice melting at room temperature. The process involves an increase in entropy – the solid (ordered phase) transforms into a liquid (disordered phase) – and absorbs heat, so enthalpy increases. However, the temperature is high enough to ensure that the entropy term TS exceeds the enthalpy increase, making ΔG negative and the process spontaneous.

Helmholtz free energy

The Helmholtz free energy, A, is defined as:

A = U – TS

where U is the internal energy. The Helmholtz free energy is most relevant in systems at constant volume and temperature.

Like the Gibbs free energy, the spontaneity of a process can be quantified by the change in the Helmholtz energy. A process is spontaneous if:

ΔA < 0

Combination of entropy and free energy

In real-world applications, both entropy and free energy play important roles in determining the thermodynamic behavior of systems. For example, in biological systems, the principle of minimizing Gibbs free energy helps predict how metabolic reactions proceed in cells.

Example of a biochemical reaction

Consider a biochemical reaction where ATP is converted to ADP. The reaction releases energy that the cell can use to do work. By calculating the Gibbs free energy change, the feasibility of the reaction can be estimated, and adjustments (e.g., changes in concentration, temperature) can optimize cellular conditions.

Visual representation

To better understand these concepts, imagine the following illustration:

initial state Mixed state: high entropy

This diagram shows the transition from an ordered state to a disordered state. The blue box represents a container of gas before the wall is removed, and the green rectangle represents its expansion afterward. The molecular expansion reflects an increase in entropy and demonstrates the natural tendency toward disorder.

Implications in the natural world

In practical terms, the laws of thermodynamics, including the factors of entropy and free energy, are evident in every aspect of life and technology. Understanding these principles can lead to innovations in efficient energy use, new materials, and solutions to environmental challenges.

Heat engines and entropy

Heat engines that convert heat energy into work are subject to the second law of thermodynamics. They are most efficient when operating between high and low temperature reservoirs, which optimizes entropy flow. The ideal efficiency is given by the Carnot efficiency, which depends on the temperature difference between the two reservoirs.

Conclusion

In short, the interrelationship between entropy and free energy governs much of the order of the universe and also explains how systems transition from one state to another. By appreciating these concepts, we gain a deeper understanding of both the natural and engineered world, leading to advances in science, technology, and the betterment of life.

Understanding and calculating entropy and free energy requires a mix of qualitative intuition and quantitative analysis. Although they may initially seem abstract, their manifestations are all around us - whether in a melting ice cube, a forest ecosystem processing sunlight, or a car engine converting gasoline into motion.


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Entropy and free energy

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