Thermodynamics

Equilibrium

Thermodynamic equilibrium is a primitive notion of the theory of thermodynamics. (A primitive notion is not defined in terms of previously defined concepts, but appears to be immediately understandable and thus taken for granted.) Other concepts of thermodynamics (transport processes, state variables) are necessarily defined upon thermodynamic equilibrium.

Thermodynamic equilibrium of a system interacting with its surroundings is the unique stable stationary state of the system that is eventually reached over a long time.

The above postulate asserts thermodynamic equilibrium as a unary relation (i.e. property) of a system. A thermodynamic system, by default, refers to a system at thermodynamic equilibrium.

Thermodynamic state

Notes on Thermodynamic state

When two systems are connected, transport processes could happen at the interface. All transport processes are primitive notions in theoretical studies of the corresponding phenomena. Specifically, heat is asserted as a distinct transport process in the theory of thermodynamics.

Transport Phenomena:

Momentum transport: viscosity;
Energy transport: heat conduction;
Mass transport: molecular diffusion;

A contact equilibrium between two systems is defined as the absence of a particular type of transport process when the systems are connected by a path permeable only to that process. Contact equilibrium shall be understood as a macroscopic equilibrium, with (almost) perfectly balanced microscopic exchanges resulting in no macroscopic flow of any kind anywhere.

Two systems are in thermodynamic equilibrium if they are simultaneously in all of the following contact equilibria:

Thermal equilibrium: No heat flow occurs between two systems when they are connected by a path permeable only to heat.
- Radiative equilibrium: No radiative heat flux between two systems when they are connected by a path permeable only to heat flux.
Mechanical equilibrium: No mechanical work occurs between two systems when they are connected by a path permeable only to mechanical work.
Chemical Equilibrium (including Phase equilibrium): No chemical reaction (including diffusion) occurs between two systems when they are connected by a path permeable only to its reactants and products†.

† Paths permeable to substance is always permeable to convective heat flow, thus chemical equilibrium needs special treatment.

The above definition establishes thermodynamic equilibrium as an equivalence relation of two systems. Since contact equilibrium can be interpreted as binary relation, by physical symmetry, every contact equilibrium is a symmetric and reflexive relation. The two contact equilibria other than thermal equilibrium have already been asserted to be Euclidean relations (similar to transitivity). Together with the Zeroth Law which asserts transitivity for thermal equilibrium, the three contact equilibria are all equivalence relations, establishing thermodynamic equilibrium as a (finer) equivalence relation.

An equivalence relation partitions a set into a collection of equivalence classes, referred to as the quotient set/space of the set by the equivalence relation. The thermodynamic equilibrium of a system is thus partitioned into a quotient set by these contact equilibria, called the (phase) space of thermodynamic states. The thermodynamic state of a thermodynamic system refers to the equivalence class of its current thermodynamic equilibrium.

State variables

Notes on State variables

If a transport process has been assigned a direction, the quotient set of equivalence classes defined by the corresponding contact equilibrium can be strictly totally ordered. To capture this order structure, we can assign a unique number to each equivalence class such that the number monotonically decreases along the direction of the transport process. The quantity such assigned can be interpreted as the driving force of the transport process.

Besides order, metric is often found useful in studying equilibrium states. In case of a one-sided chain with metric, the structure of the quotient set is typically modeled as nonnegative real numbers for convenience.

This type of quantities are intensive properties of a system, because subdividing the system does not affect their value: any two subsystems of a thermodynamic system are in thermodynamic equilibrium with each other. Temperature is the intensive property of a system that drives heat transfer.

Every macroscopic intensive property of a thermodynamic system is balanced. In absence of other factors triggering a transport process, the property corresponding to the process must be spatially uniform in a system at equilibrium. Intensive properties other than temperature may be driven to spatial inhomogeneity by a static long-range force field imposed on it by its surroundings. Such equilibrium inhomogeneity does not occur to temperature because, borrowing primitive notions from mechanics, long-range "force" that drives heat flow is not necessary in thermodynamics.

With one macroscopic intensive property for each transport process, we can identify the thermodynamic states of a system with an \(n\)-tuple of nonnegative numbers: pressure, temperature, and chemical potentials. But not all of them are independent; they are related by the Gibbs–Duhem equation in each phase. (See Chemical Equilibrium)

A definite number of real variables define the states that are the points of the manifold of equilibria. [Carathéodory, C. (1909)]

The dimension of the space of thermodynamic system is determined by Gibbs' phase rule: For a non-reactive thermodynamic system involving \(c\) components (组分) and \(p\) phases (\(c \ge p\)), the degree of freedom of its thermodynamic equilibria is \(2+c-p\). For example, the space of thermodynamic states of a single-species single-phase system is two dimensional, where the two corresponds to the mechanical and thermal transport processes. Reasoning of Gibbs' phase rule: System at thermodynamic equilibrium has uniform temperature and pressure. If a system consists of multiple components, since it must be homogeneous in each phase at equilibrium, only the amount fractions in each phase count as additional degrees of freedom.

A state variable of a thermodynamic system is a property of the system that only depends on the current thermodynamic equilibrium. A state function, with subtle difference, refers to a state variable regarded as a function of other state variables. Obviously, the intensive properties for transport processes are state variables. State variables can also be extensive properties: volume, entropy, amount of a chemical species. Essentially any function of the thermodynamic states of a system is its state variable.

Topics

Statistical Distributions
- Maxwell distribution for velocity and speed
- Boltzmann distribution for number density
- Maxwell–Boltzmann distribution for energy
- Equipartition theorem & heat capacity
Phase Transition

Dynamics

Laws of thermodynamics

Zeroth law of Thermodynamics:

Thermal equilibrium between two systems is a transitive relation. [Carathéodory, C.]

All heat is of the same kind. [Maxwell, J.C. (1871), p. 57.]

First law of Thermodynamics (for closed and open systems)

Heat is the transfer of (internal) energy.

Second law of Thermodynamics

Heat can never autonomously transfer along the temperature gradient.

Third law of Thermodynamics

As the temperature of any condensed system approaches absolute zero, its entropy change in any process also approaches zero.

Only the second and the third laws are actually empirical laws. In fact, Constantin Carathéodory (1909) formulated thermodynamics, up to the second law, on a purely mathematical axiomatic foundation.

Major implications of the laws of thermodynamics:

The Zeroth Law of Thermodynamics establishes temperature as one-dimensional, unlike one chemical potential for each component.
The First Law of Thermodynamics establishes heat as the transfer of a unique form of energy, which thus can transform with other forms of energy but conserves as whole in isolated systems.
The Second Law of Thermodynamics justifies entropy as a state variable of a system (integral of the ratio of incremental heat transfer divided by temperature in a reversible process).
The Third Law of Thermodynamics justifies (Planck) absolute entropy.

First Law and thermodynamic potentials

The First Law effectively unifies distinct physicochemical processes under the umbrella of energy, a conserved quantity in each transport phenomenon. The First Law states that, when the three types of transport processes of thermodynamics are considered, the change of internal energy of a system equals the sum of mechanical work done to the system by the environment, heat flowing into the system, and energy added by new particles: (\(\delta\) denotes changes of process functions, \(\mathrm{d}\) denotes changes of state functions.)

\[ \mathrm{d} U = - \delta W + \delta Q +\sum_i \mu_i\,\mathrm{d}N_i \]

Starting from internal energy, different thermodynamic potentials can be derived by (non-standard) Legendre transforms, each time substituting a independent state variable with another state variable. The two state variables form a conjugate pair, a pair of intensive and extensive properties whose product refers to action/energy, and are analogous to generalized force and generalized coordinate in classical mechanics. Either state variable of the pair is called the conjugate variable of the other.

Table: State Variables as Conjugate Pairs of Action by Transport Process

Process	Action	Intensive	Extensive
Mechanical	\(W\)	\(p\)	\(V\)
Thermal	\(Q\)	\(T\)	\(S\)
Material	\(G\)	\({μ_i}\)	\({N_i}\)

The Legendre transforms are nonstandard in that it is the difference, rather than the sum, of the two functions that equals the product of the conjugate pair.

\[ \begin{align} U(V, S, \{N_i\}) - H(p, S, \{N_i\}) &= -p V \\ U(V, S, \{N_i\}) - F(V, T, \{N_i\}) &= T S \\ F(V, T, \{N_i\}) - G(p, T, \{N_i\}) &= -p V \\ H(p, S, \{N_i\}) - G(p, T, \{N_i\}) &= T S \\ F(V, T, \{N_i\}) - \Psi(V, T, \{\mu_i\}) &= \sum_i \mu_i N_i \end{align}\]

This strategy is used to shift each of the independent state variables between extensive and intensive properties, so that one thermodynamic potential will be the easiest to work with and thus the most useful, depending on the controlled variables in a specific situation. For example, Gibbs free energy is the most useful thermodynamic potential in studies of chemical reactions, because most reactions happen at constant pressure and temperature. For a space of thermodynamic states with \(D\) dimensions, there are \(2^D\) unique thermodynamic potentials.

The complete set of thermodynamic (energy) potentials in closed systems and their interpretations:

Internal energy \(U\): is the capacity to do mechanical work (useful work) plus the capacity to release heat.
Enthalpy \(H\): the capacity to do non-mechanical work plus the capacity to release heat.
Helmholtz free energy \(F\): the capacity to do mechanical work.
Gibbs free energy \(G\): the capacity to do non-mechanical work.

A closed system has no material exchange with the environment, while an open system is the opposite. A common thermodynamic potential in open systems is the grand potential (Landau Potential) \( \Psi \). All the common thermodynamic potentials are energy potentials, but there are also entropy potentials, know as free entropies.

Table: Relations among thermodynamic potentials of a closed system

Capacity	mechanical work, \(-pV\)	no mechanical work
heat, \(TS\)	\(U\)	\(H\)
no heat	\(F\)	\(G\)

First order partial derivatives of thermodynamic potentials with respect to their independent variables are the conjugate variables of the independent variable.

\[ p_i = \frac{\partial \Phi}{\partial q_i} \]

These relations are called equations of state, if the partial derivatives are expressed as functions of state variables. All equations of state for a thermodynamic potential forms the constitutive relation of a given system. If the space of thermodynamic states of a system has \(D\) dimensions, it has \(D\) constitutive relations mapping between the independent conjugate pairs. You need one additional extensive state variable (e.g. total particle number or volume) to fully specify its thermodynamic property.

Second partial derivatives of thermodynamic potentials are material properties: coefficient of thermal expansion, coefficient of thermal pressure; isothermal compressibility, adiabatic compressibility; specific heat (at constant pressure, at constant volume). Material properties are related by Maxwell relations: mixed second order derivatives are equal.

\[ \frac{\partial^2 \Phi}{\partial q_i \partial q_j} = \frac{\partial^2 \Phi}{\partial q_j \partial q_i} \]

Second Law and its implications

Clausius inequality as the Second Law:

\[ \mathrm{d}S \ge \frac{\delta Q}{T} \]

The Second Law has many implications:

implies the entropy of a system never decrease in adiabatic processes.
implies isolated systems always spontaneously increase their entropy.
implies Carnot's theorem (maximum efficiency of heat engines).
justifies an absolute temperature scale.

A quasi-static process is a process throughout which the system is always in thermodynamic equilibrium. A reversible process is a quasi-static process that causes no change in the total entropy of the system and its surroundings. Note that reversible processes are not necessarily isentropic for the system. Transport phenomena are all irreversible.

Principle of maximum work:

In isothermal processes, the maximum work a system can do to the environment equals the decrease of its Helmholtz free energy.
In isothermal isobaric processes, the maximum non-mechanical work a system can do equals the decrease of its Gibbs free energy.

For all thermodynamic processes between the same initial and final states, the delivery of work is a maximum for a reversible process.

Criteria for thermodynamic equilibrium of a closed system: (principle of maximum entropy; principle of minimum energy)

If isolated (isovolumetric, constant energy), a thermodynamic system has the maximum entropy.
In isentropic isovolumetric conditions, a thermodynamic system has the minimum internal energy.
In isothermal isovolumetric conditions, a thermodynamic system has the minimum Helmholtz free energy.
In isothermal isobaric conditions, a thermodynamic system has the minimum Gibbs free energy.

Dynamic relations

Notes on Dynamic relations

All properties of the thermodynamic states of a system can be determined with the fundamental thermodynamic relation (or its Euler integral) and the system's constitutive relations. The fundamental thermodynamic relation is based on physical laws, while constitutive relations are phenomenological, unless derivable from statistical mechanics.

The fundamental thermodynamic relation is first order equation of a thermodynamic potential, which combines the First and the Second Law (for reversible processes). Natural variables for a thermodynamic potential refers to the state variables appeared as differentials in the corresponding fundamental thermodynamic relation. The fundamental thermodynamic relation is one equation, which may be expressed with different natural variables and corresponding thermodynamic potentials:

\[ \begin{align} \mathrm{d} U &= &T \mathrm{d} S - p \mathrm{d} V + \sum_i \mu_i \mathrm{d} N_i \\ \mathrm{d} H &= &T \mathrm{d} S + V \mathrm{d} p + \sum_i \mu_i \mathrm{d} N_i \\ \mathrm{d} F &= &-S \mathrm{d} T - p \mathrm{d} V + \sum_i \mu_i \mathrm{d} N_i \\ \mathrm{d} G &= &-S \mathrm{d} T + V \mathrm{d} p + \sum_i \mu_i \mathrm{d} N_i \\ \mathrm{d} \Psi &= &-S \mathrm{d} T - p \mathrm{d} V - \sum_i N_i \mathrm{d} \mu_i \end{align}\]

Euler integrals of the fundamental thermodynamic relations determine thermodynamic potentials up to a reference constant. With all natural variables being extensive properties, internal energy is a homogeneous function of degree 1. Following Euler's homogeneous function theorem, the Euler integral of internal energy is: \[ U = T S - p V + \sum_i \mu_i N_i \]

Once a thermodynamic potential is determined as a function of its natural variables, all the system's thermodynamic properties can be found as partial derivatives of that thermodynamic potential.

Statistical Thermodynamics

Statistical thermodynamics, aka equilibrium statistical mechanics, studies the properties of thermodynamic states of a system of particles. It derives state variables and equations of state from ensemble's probability distribution over micro-states. If not static, the ensemble evolution is given by the Liouville equation for classical particles, which can be derived from particle equations of motion, which in this case are Hamilton's equations. Statistical thermodynamics also studies further the into microscopic level, such as fluctuations. In other words, it takes thermodynamics from phenomenology to mechanistic theory.

Statistical ensemble theory

Notes on Statistical ensemble theory

A statistical ensemble is a massive collection of independent hypothetical systems; each system is a massive collection of equivalent particles and its configuration evolves under a same set of dynamic equations.

Three scales of statistical ensemble model: (micro) particle -> (macro) system -> ensemble.

An ensemble is at statistical equilibrium if for each micro-state in it, the ensemble also contains all the future and past micro-states with equal probabilities.

The fundamental postulates of statistical mechanics: equal a priori probability postulate for isolated systems.

Ergodic hypothesis. [most systems are not ergodic.]
Principle of indifference.
Maximum entropy: the correct ensemble has the largest Gibbs entropy (probability).

Mean values of element states in a system are equal to mean values of corresponding system states in an ensemble.

Note Outline:

Ensemble theory
1. Fundamental concepts
2. Microstate
Three equilibrium ensembles
1. Micro-canonical ensemble
2. Canonical ensemble
3. Grand canonical ensemble

Boltzmann's entropy formula: A macroscopic state of a system is a distribution on the microstates. Entropy is a measure of this distribution.

Topics

Topics in statistical mechanics

Systems of approximate non-interacting particles
1. distribution over energy level derived from ensemble theory
2. most probable distribution over energy level
Quasi-thermodynamic theory of Fluctuations (probability distribution)

Non-equilibrium thermodynamics

Local (Planckian) temperature of a system can be defined such that emission is given by Kirchhoff's law. A system is in local thermodynamic equilibrium if the local kinetic (Maxwellian) temperature is equal to the Planckian temperature of the radiation field. In a system at local thermodynamic equilibrium, the thermodynamic driving forces are varying in space and time so slowly that one can assume thermodynamic equilibrium in some neighborhood about any point in the system. For example, temperature in a cup of water with a melting ice cube.

Onsager reciprocal relations for transport processes of near-equilibrium systems: Linearly relating the gradient of entropic forces and the rate of entropy production due to irreversible processes, the Onsager matrix of kinetic coefficients is positive semi-definite. {Onsager1931}

Conjecture of minimum entropy production: At near-equilibrium stationary states, a system under purely linear diffusion with negligible inertia has pointwise minimum rate of entropy production. {Prigogine1945}

Dissipative structures of systems far from thermodynamic equilibrium: importation and dissipation of energy (and matter) could result in dynamical regimes that can be regarded as thermodynamic steady states in a system. {Prigogine1967}

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