The reduction of thermodynamics to

mechanics

Bryan W Roberts, LSE

IHPST/SPHere
These slides: personal.lse.ac.uk/robert49/talks/paris
Paper: arxiv:2503.08753
Join: philosophyofphysics.org

Foundations of thermodynamics

The central problem:

Thermodynamics is successful but seems to be strictly false.

• Fundamental physics: Fluctuations, time symmetry
• Thermodynamics: No fluctuations, (apparently) no time symmetry

The central problem:

Philosophical Responses

The central problem:

A New Response

Shared Structure: Thermodynamics and mechanics retain some autonomy but also happen to agree because of their shared structure.
(BWR 2023: 'European' approach)

What is the shared structure?
Thermodynamics is identical to mechanics
when some energy, which we call 'heat', unobservable.

Talk plan

1. The basic idea
2. Reduction and contact geometry
3. The arrow of time

arxiv:2503.08753

The trouble with physics

The trouble with physics

General Relativity

The world is a Lorentzian manifold \((M,g_{ab})\)

The trouble with physics

Quantum Theory

The world is a Hilbert space \((H,\mathcal{A})\)

The trouble with physics

Gauge theory

The world is a vector bundle with a connection \((V,\nabla)\)

The trouble with physics

Thermodynamics

"There is a semi-permeable membrane in contact with a heat bath."

????

Thermodynamics

"Every mathematician knows it is impossible to understand an elementary course in thermodynamics. ...The reason is that the thermodynamics is based — as Gibbs has explicitly proclaimed — on a rather complicated mathematical theory, on the contact geometry." Arnol'd (1990, p.163) → cf. Uffink 2001, p.310)

Thermodynamics

A new proposal

The world is a mechanical system with some observables, \((M,\omega,S)\)
and where some of the energy is unobservable.

\(dh = PdV\)

\(\xi = \) the heat one form (\(= \text{đ}Q\))

Phase space: \((T^*C,\Omega)\)

Phase space: \((T^*C,\Omega)\)

Energy

\(\rho = -p_1dq_1 - p_2dq_2 - p_3dq_3 - p_4dq_4 - p_5dq_5 - \cdots - p_{10^{23}}dq_{10^{23}}\)

Phase space: \((T^*C,\Omega)\)

Energy

\(\rho = -p_1dq_1 - p_2dq_2 - p_3dq_3 - p_4dq_4 - p_5dq_5 - \cdots - p_{10^{23}}dq_{10^{23}}\)

Measurement procedures

Phase space: \((T^*C,\Omega)\)

Energy

\(\rho = -\)\(\underbrace{PdV}_{\text{measurable}}\)\( - \underbrace{p_2dq_2 - p_3dq_3 - p_4dq_4 - p_5dq_5 - \cdots - p_{10^{23}}dq_{10^{23}}}_{\text{non-measurable}}\)

Measurement procedures

Phase space: \((T^*C,\Omega)\)

Energy

\(dU = -\)\(\underbrace{\omega}_{PdV}\)\( - \underbrace{p_2dq_2 - p_3dq_3 - p_4dq_4 - p_5dq_5 - \cdots - p_{10^{23}}dq_{10^{23}}}_{\text{non-measurable}}\)

Measurement procedures

Phase space: \((T^*C,\Omega)\)

Energy

\(dU = -\)\(\underbrace{\omega}_{PdV}\)\( + \underbrace{\xi}_{TdS}\)

Measurement procedures

Phase space: \((T^*C,\Omega)\)

Energy

\(dU = -\)\(\underbrace{\omega}_{PdV}\)\( + \underbrace{\xi}_{TdS}\)

First Law of Thermodynamics

\(\xi\) \(= dU +\) \(\omega\)

"Decomposition of energy into observable vs. hidden parts"

First Law of Thermodynamics

\(\xi\) \(= dU +\) \(\omega\)

"Decomposition of energy into observable vs. hidden parts"

Some special cases:

• Heat = microscopic (doesn't apply to black holes)
• Heat = unknown information (requires a knower)
• Heat = uncontrolled variables (Wallace 2014, Myrvold 2020)

Some more technical material

Some more technical material

What is thermodynamic state space?

The arrow of time

What is thermodynamic state space?

What is thermodynamic state space?

Mechanical state space

\(dU = -\)\(\underbrace{\omega}_{PdV}\)\( - \underbrace{p_2dq_2 - p_3dq_3 - p_4dq_4 - p_5dq_5 - \cdots - p_{10^{23}}dq_{10^{23}}}_{\text{non-measurable}}\)

What is thermodynamic state space?

Thermodynamics

\(dU +\) \(\omega\) \(=\) \(\xi\) \(= TdS\)

What is thermodynamic state space?

Answer: The reduction of many hidden degrees of freedom to just one called entropy \(S\), with temperature \(T\) its conjugate.

What is thermodynamic state space?

\(\xi\) \(= dU +\) \(\omega\)

'Reduction' of a dynamical system \((T^*C,\Omega)\) to a contact system \((M,\theta,L)\)

'Reduction' of a dynamical system \((T^*C,\Omega)\) to a contact system \((M,\theta,L)\)

• \(\dim M=2n+1\), local coordinates \((P_1,V_1,...,P_{n-1},V_{n-1},T,S,U)\)
• Contact form: \(\theta = dU - TdS - \sum_{i=1}^n P_idV_i\)
• \(\xi := \theta + TdS\) and \(\omega = - \sum_{i=1}^n P_idV_i\):
  \(\xi = dU + \omega\)
• \(\Rightarrow\) First law: \(\xi = dU + \omega\)
  
  
• \(T_pL=\ker\theta\big|_p\), i.e. \(\theta\big|_L = 0\) \(\Leftrightarrow\) \(\xi = TdS\) on \(L\)

'Reduction' of a dynamical system \((T^*C,\Omega)\) to a contact system \((M,\theta,L)\)

• \(\dim M=2n+1\), local coordinates \((P_1,V_1,...,P_{n-1},V_{n-1},T,S,U)\)
• Contact form: \(\theta = dU - TdS - \sum_{i=1}^n P_idV_i\)
• \(T_pL=\ker\theta\big|_p\), i.e. \(\theta\big|_L = 0\) \(\Leftrightarrow\) \(\xi = TdS\) on \(L\)
• \(\dim L = n\), so one can choose any \(n\), and the remaining will be functions.
• Ex: if \((S,V_1,\dots,V_{n-1})\) are coordinates, then \(U,T,P_1,...,P_{n-1}\) are functions
• These define equations of state e.g. \(P_i = \partial U/\partial V_i\), \(T = \partial U/\partial S\), etc.

'Reduction' of a dynamical system \((T^*C,\Omega)\) to a contact system \((M,\theta,L)\)

Contact geometry is the deep structure of thermodynamics:

• \((2n+1)\)-dimensional manifold \(M\)
• One-form \(\theta\) defining a field of hyperplanes
• \(\theta\) is a contact form: \((d\theta)^n\wedge\theta \neq 0\).
• Legendre submanifold \(\varphi:L\rightarrow M\) with \(\varphi^*\theta = 0\) and \(\text{dim}\, L=n\).
• \(\Rightarrow\) Precise expression of the laws, Maxwell relations, phase transitions...

Reduction Procedure: BWR 2025

The arrow of time

The arrow of time

• For us, thermodynamics \(=\) mechanics with some unobservable energy.
• So, their arrows of time align, and the mystery is dissolved.
  
  
• But what about the entropy arrow?

The arrow of time

The thermal entropy arrow is non-conservative.

• And, arrows that are non-conservative are no mystery.
• They occur in mechanics in the same way.
• So, the mystery is dissolved.

The thermal entropy arrow is non-conservative.

Clausius Entropy Argument: if a function \(T\) and a curve \(\gamma\) from \(p_i\) to \(p_f\) satisfy certain assumptions...

• \(\gamma\) is an adiabat
• there is a curve \(\lambda\) from \(p_f\) to \(p_i\) on which \(\xi = TdS\) for some \(S\)
• Clausius principle \(\int_c \xi/T \leq 0\)

...then \(S(p_i) \leq S(p_f)\).

Loophole: No time asymmetry if \(S(p_i)=S(p_f)\).

Claim: if \(S(p_i) \neq S(p_f)\), then energy conservation is violated.

The thermal entropy arrow is non-conservative.

Claim: if \(S(p_i) \neq S(p_f)\), then energy conservation is violated.

• Exercise: If \(S(p_i)\neq S(p_f)\), then there is a curve on which \(\xi\neq TdS\).
• Afanassjewa-Jauch Theorem: If \(\xi\neq TdS\), then energy conservation is violated (i.e. work is not conserved on adiabats).

Summary. If Clausius entropy increases, then energy conservation is violated.

The arrow of time

The thermal entropy arrow is non-conservative.

Summary. If Clausius entropy increases, then energy conservation is violated.

The arrows of thermodynamics and mechanics align.

A New Response

Shared Structure: Thermodynamics and mechanics retain some autonomy but also happen to agree because of their shared structure.
(BWR 2023: 'European' approach)

What is the shared structure?
Thermodynamics is identical to mechanics
when some energy, which we call 'heat', unobservable.

Appendix

Afanassjewa-Jauch Theorem

Theorem. Let \(\pi:L\rightarrow N\) be a line bundle, \(U\) a vertical coordinate function, and \(\omega = \sum_{i=1}^{n-1}P_i dV_i\) a one-form on \(L\) with \(V_i\) coordinates in \(N\), so \(\ker\pi_*\subseteq\ker\omega\). Let \(\xi := dU + \omega\) be nowhere-vanishing. If,

(Energy conservation for work-closed adiabats): for every piecewise-smooth \(\gamma\) in \(L\), if \(\pi[\gamma]\) is closed and \(\xi(\bar{\gamma})=0\), then \(\int_\gamma\omega=0\),

then \(\xi = TdS\) for some smooth \(T,S\) in a neighbourhood of each \(p\in L\).