Pith Number

pith:WG7H7ULM

pith:2026:WG7H7ULMSQ7UZ2TBLNM4K6F2YL

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Brownian motion: non-equilibrium states from equilibrium trajectories -- recovering hydrodynamic regimes from prepared displacement measurements

Giuseppe Procopio, Jason Boynewicz, Massimiliano Giona, Michael C. Thumann

Any equilibrium Brownian trajectory decomposes into a superposition of non-equilibrium states via the Chapman-Kolmogorov equation.

arxiv:2605.16247 v1 · 2026-05-15 · cond-mat.stat-mech

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Claims

C1strongest claim

Owing to the Chapman-Kolmogorov equation for Markovian dynamics, any equilibrium trajectory of a Brownian particle in a solvent fluid can be viewed as the superposition of an uncountable number of non-equilibrium states, permitting the unraveling of fine details of fluid-particle interactions at microscales from the analysis of a single Brownian trajectory by considering the lower-order (second) moments of particle position in trapped conditions.

C2weakest assumption

The particle dynamics are strictly Markovian so that the Chapman-Kolmogorov equation applies directly to the equilibrium trajectory and allows its decomposition into non-equilibrium components (stated in the opening sentence of the abstract).

C3one line summary

Equilibrium Brownian trajectories encode non-equilibrium hydrodynamic information through displacement moments, confirming a t^{5/2} scaling from fluid inertia and suggesting a possible t^4 scaling at shorter times due to velocity regularity.

References

63 extracted · 63 resolved · 0 Pith anchors

[1] 001 0 . 01 0 . 1 1 10 100 1000 mxx(t) 1 10 t5 FIG. 2. mxx(t) = ⟨x2(t)|v(0) = 0 , R (0) = 0 ⟩ vs t expressed by eq. (35) for zero velocity and zero initial thermal force conditions for a GLE governed b

[2] ﬁll the plane

[3] Chandrasekhar, Stochastic Problems in Physics and As tronomy, Rev 1943

[4] Einstein, ¨Uber die von der molekularkinetischen Theorie der W¨ rme gef orderte Bewegung von in ruhenden Fl¨ ussigkeiten suspendierten Teilchen,Ann 1905

[5] Einstein, Zur Theorie der Brownschen Bewegung Ann 1906

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:02:00.036431Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b1be7fd16c943f4cea615b59c578bac2cbde4410393e80ec2f4e2ed9b58b6b01

Aliases

arxiv: 2605.16247 · arxiv_version: 2605.16247v1 · doi: 10.48550/arxiv.2605.16247 · pith_short_12: WG7H7ULMSQ7U · pith_short_16: WG7H7ULMSQ7UZ2TB · pith_short_8: WG7H7ULM

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WG7H7ULMSQ7UZ2TBLNM4K6F2YL \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b1be7fd16c943f4cea615b59c578bac2cbde4410393e80ec2f4e2ed9b58b6b01

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d6edd67a7f5f5d34543da68ddfe1312af88cbc6846d583c9e0f56e44587d8fff",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cond-mat.stat-mech",
    "submitted_at": "2026-05-15T17:51:01Z",
    "title_canon_sha256": "3b2c734d6be43247ae329e10d1c9a9f99b6727664cc243f4d7247a50dbf7de17"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16247",
    "kind": "arxiv",
    "version": 1
  }
}