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Josef Loschmidt (1821–1895)

Loschmidt posed the sharpest early objection to Boltzmann’s statistical mechanics — now standardly called the Loschmidt paradox or the reversibility paradox. If the microscopic laws of motion are time-symmetric — for every trajectory from low entropy to high entropy, there exists an equally valid time-reversed trajectory from high to low — then no time-asymmetric conclusion (entropy always increases) can follow from time-symmetric premises. The objection, raised in 1876, forced Boltzmann to clarify that his H-theorem established the overwhelming probability of entropy increase, not its logical necessity — a clarification that sharpened the statistical interpretation of the second law and revealed that the arrow of time is not a consequence of the laws of mechanics alone but of initial conditions. Loschmidt also made the first reliable estimate of molecular size and the number of molecules in a given volume of gas — “Loschmidt’s number” — a foundational contribution to the atomic hypothesis.

Life

Born 15 March 1821 in Putschirn (Počerny), Bohemia (now Czech Republic), into a poor farming family. Studied at the Charles University in Prague and the Vienna Polytechnic Institute, but his early career was marked by financial difficulty and failed business ventures — including a potassium nitrate factory that collapsed during the 1848 revolution. He worked as a schoolteacher in Vienna for over a decade before entering academic science late.

Appointed Privatdozent at the University of Vienna (1866), then extraordinary professor (1868), then full professor of physical chemistry (1875). The academic career began when Loschmidt was in his mid-forties — late by any standard. His most productive period was the 1860s and 1870s.

Loschmidt and Boltzmann were colleagues and friends at the University of Vienna. The reversibility objection was raised in a spirit of scientific engagement, not hostility — Loschmidt admired Boltzmann’s work and sought to clarify its foundations, not to destroy it. Boltzmann took the objection seriously and responded substantively. Died 8 July 1895 in Vienna, the year before Boltzmann’s other great critic, Ernst Zermelo, raised the recurrence objection.

Molecular size and Loschmidt’s number

In 1865, Loschmidt published the first reliable estimate of the size of molecules in air. Using the kinetic theory of gases (the mean free path of molecules, as developed by Clausius and Maxwell), together with data on the density and viscosity of air, he calculated that a molecule of air has a diameter of approximately one nanometre — remarkably close to modern values. From this he derived the number of molecules in a cubic centimetre of gas at standard temperature and pressure: approximately 2.6 × 10¹⁹ — now called Loschmidt’s number (N_L). The related quantity — the number of molecules in a mole of substance — is Avogadro’s number (N_A ≈ 6.022 × 10²³), which in German-speaking countries was long called “Loschmidt’s number” as well.

The estimate was significant because it provided concrete physical evidence for the atomic hypothesis at a time when many physicists — including Ernst Mach and the energeticists — denied that atoms were real entities. Loschmidt’s calculation gave atoms a size and a number: they were not merely useful fictions but objects whose properties could be measured.

The reversibility objection

Loschmidt’s objection (1876) to Boltzmann’s H-theorem runs as follows:

The H-theorem purports to show that entropy increases irreversibly — that a gas in a non-equilibrium state will approach equilibrium and remain there. But the underlying mechanics (Newtonian or Hamiltonian) is time-reversible: if every particle’s velocity is reversed at a given instant, the system retraces its trajectory exactly, moving from higher entropy back to lower entropy. For every entropy-increasing trajectory, there exists a time-reversed entropy-decreasing trajectory that is equally consistent with the laws of mechanics. How can a time-asymmetric conclusion (the second law) follow from time-symmetric premises (the laws of motion)?

Boltzmann’s response. The H-theorem does not prove that entropy must increase for every microstate; it proves that entropy increase is overwhelmingly probable. The time-reversed trajectories exist but are astronomically improbable — they require initial conditions of extraordinary precision (every velocity reversed exactly). A randomly chosen initial condition will, with overwhelming probability, evolve toward higher entropy. The arrow of time, on this account, is not built into the laws of mechanics but into the initial conditions: the universe began (or the system was prepared) in a low-entropy state, and the statistical mechanics does the rest.

The exchange between Loschmidt and Boltzmann is one of the foundational episodes in the philosophy of statistical mechanics. It established that the second law is statistical, not absolute; that the arrow of time requires an explanation beyond the dynamical laws; and that the relationship between microscopic reversibility and macroscopic irreversibility is a genuine conceptual problem, not a confusion that disappears once the mathematics is done correctly.

Where Loschmidt stops

Loschmidt raised the reversibility objection but did not develop an alternative theory of irreversibility. The question he posed — why does the universe have a low-entropy initial condition? — remains open. Modern treatments (Roger Penrose’s Weyl curvature hypothesis, David Albert’s Past Hypothesis, Sean Carroll’s cosmological approach) all attempt to answer the question Loschmidt identified, but none has achieved consensus. The reversibility objection is more productive as a question than as an argument: it does not refute Boltzmann’s statistical mechanics but reveals the assumptions on which the entire framework rests.

Loschmidt’s structural chemistry work — he proposed molecular structures for organic compounds as early as 1861, independently of August Kekulé — has been largely eclipsed in the historical record by Kekulé’s priority claims. Part of the explanation is practical: Chemische Studien I was published by Loschmidt’s employer Carl Gerold in a small print run of around 250 copies and did not circulate widely, while Kekulé’s later publications reached a broad audience through established journals. Whether Loschmidt’s structural diagrams anticipated Kekulé’s benzene ring and related structures, or whether they were different in kind, is a question in the history of chemistry that has been debated but not fully resolved — but the limited circulation of the 1861 work is part of why the question remained unresolved for so long.

Key works

See also: Boltzmann · Maxwell · Penrose