BioSys ems
Maxwell’s Demon in a High-Dimensional Uni e se Cohe ence Collapse and he Limi s
o Landaue E asu e
--Manusc ip D a --
Manusc ip Numbe :
A icle Type: Full Leng h A icle
Sec ion/Ca ego y:
Keywo ds: Maxwell's demon; Landaue p inciple; high-dimensional dynamics; cohe ence;
dimensional collapse; in o ma ion he modynamics; sub-Landaue e asu e; quan um
biology
Co esponding Au ho : Ian Todd
Uni e si y o Sydney
AUSTRALIA
Fi s Au ho : Ian Todd
O de o Au ho s: Ian Todd
Abs ac : We gene alize Landaue ’s p inciple o high-dimensional dynamical subs a es whe e
in o ma ion is embodied as cohe ence geome y a he han disc e e bi s. In his
se ing, measu emen and e asu e a e dimensional collapses—coa se-g ained
p ojec ions om a high-dimensional mani old on o a lowe -dimensional a ac o . We
de i e acohe ence-co ec ed bound Ee ase, educing o kBT ln2 o bina y con ac ion.
The demon’s ”memo y” is in e nal co ela ion (an a ac o ) wi hin he same ield, so
in o ma ion acquisi ion is cohe ence o ma ion and e asu e is decohe ence. Because
collapse in one subspace can be o se by expansion in ano he , local appa en cos s
can all below kBT ln2 while he global second law is p ese ed. This in o ma ion-
geome ic iew uni ies Maxwell’s demon, Landaue ’s limi , and biological/analog
compu a ion, and yields es able scaling wi h los dimensionali y and cohe ence.
Powe ed by Edi o ial Manage ® and P oduXion Manage ® om A ies Sys ems Co po a ion
Ian Todd
Sydney Medical School
Uni e si y o Sydney
Sydney, NSW, Aus alia
i o[email p o ec ed].edu.au
Oc obe 10, 2025
D . Abi Igambe die
Edi o -in-Chie
BioSys ems
Dea D . Igambe die ,
Please ind enclosed ou manusc ip , “Maxwell’s Demon in a High-Dimensional Uni e se: Cohe ence
Collapse and he Limi s o Landaue E asu e,” submi ed o BioSys ems as a Regula A icle
(Theo y and Modelling).
This wo k gene alizes Landaue ’s p inciple o high-dimensional cohe ence dynamics by ea ing
e asu e as dimensional collapse. We de i e a cohe ence-co ec ed bound
Ee ase ≥kBTe lnV(De )
V(D′)
ha educes o kBTln 2 o bina y sys ems and explains how appa en sub-Landaue e asu e a ises
locally h ough dimensional edis ibu ion while espec ing global he modynamic cons ain s. We
ein e p e Maxwell’s demon as an in e nal a ac o , esol ing he ex e nal-obse e pa adox. The
amewo k yields es able p edic ions o dimensional scaling, cohe ence dependence, and me abolic
e iciency. A nume ical appendix alida es he key claims; simula ion code is embedded and epli-
cable wi h NumPy.
This pape is concep ually dis inc om my ecen BioSys ems a icle on alsi iabili y limi s, p o-
iding he he modynamic ounda ions and geome ic bound ha complemen he ea lie epis e-
mological pe spec i e.
The manusc ip is o iginal and no unde e iew elsewhe e. Du ing p epa a ion I used Claude
(An h opic) o d a ing, wi h Cha GPT and G ok o e i ica ion. All esul s we e alida ed by
me. I ha e no compe ing in e es s.
Since ely,
Ian Todd
Sydney Medical School
Uni e si y o Sydney
Co e Le e
Maxwell’s Demon in a High-Dimensional Uni e se:
Cohe ence Collapse and he Limi s o Landaue E asu e
Ian Todd
Sydney Medical School
Uni e si y o Sydney
Sydney, NSW, Aus alia
[email p o ec ed]
Oc obe 10, 2025
Abs ac
We gene alize Landaue ’s p inciple o high-dimensional dynamical subs a es whe e
in o ma ion is embodied as cohe ence geome y a he han disc e e bi s. In his se -
ing, measu emen and e asu e a e dimensional collapses—coa se-g ained p ojec ions
om a high-dimensional mani old on o a lowe -dimensional a ac o . We de i e a
cohe ence-co ec ed bound Ee ase ≥kBTe ln V(De )/V (D′), educing o kBTln 2
o bina y con ac ion. The demon’s ”memo y” is in e nal co ela ion (an a ac o )
wi hin he same ield, so in o ma ion acquisi ion is cohe ence o ma ion and e asu e is
decohe ence. Because collapse in one subspace can be o se by expansion in ano he ,
local appa en cos s can all below kBTln 2 while he global second law is p ese ed.
This in o ma ion-geome ic iew uni ies Maxwell’s demon, Landaue ’s limi , and bio-
logical/analog compu a ion, and yields es able scaling wi h los dimensionali y and
cohe ence.
1
Manusc ip File Click he e o iew linked Re e ences
Keywo ds: Maxwell’s demon, Landaue p inciple, high-dimensional dynamics, cohe -
ence, dimensional collapse, in o ma ion he modynamics, sub-Landaue e asu e, quan um
biology
1 In oduc ion
Maxwell’s demon, concei ed in 1867 as a hough expe imen challenging he second law
o he modynamics, con inues o illumina e he deep connec ion be ween in o ma ion and
physical en opy [1, 2]. The demon—a hypo he ical en i y capable o obse ing indi idual
molecula eloci ies—ope a es a apdoo be ween wo chambe s, selec i ely allowing as
molecules o pass one way and slow molecules he o he , he eby c ea ing a empe a u e
di e ence wi hou pe o ming wo k. This appa en iola ion o he second law oubled
physicis s un il Landaue ’s seminal insigh [3]: he demon mus e en ually e ase i s memo y
o ope a e cyclically, and his e asu e necessi a es ene gy dissipa ion o a leas kBTln 2 pe
bi , es o ing he modynamic consis ency.
Landaue ’s p inciple has p o en ema kably obus , alida ed expe imen ally [4, 5] and
ex ended o quan um [6, 7, 8, 9] and ela i is ic [10] domains. The amewo k has been
u he de eloped h ough eedback con ol [11, 12] and au onomous demon implemen a ions
[13], wi h comp ehensi e e iews es ablishing in o ma ion he modynamics as a ma u e ield
[14]. Ye i s s anda d o mula ion embeds assump ions ha become p oblema ic o complex
biological and physical sys ems ope a ing in high-dimensional phase spaces wi h cohe ence
dynamics. Speci ically:
1. Disc e e s a e space: The bi model assumes well-de ined, sepa able bina y s a es
a he han con inuous mani olds.
2. Low-dimensional phase space: Con en ional ea men s neglec co ela ions and
hidden deg ees o eedom ha domina e biological in o ma ion p ocessing.
2
3. The mal equilib ium: The s anda d de i a ion p esumes a uni o m hea ba h a
empe a u e T, inadequa e o sys ems wi h s uc u ed s ochas ic esonance o a -
om-equilib ium dynamics.
4. Ex e nal demon: The demon is ea ed as sepa a e om he sys em i obse es,
igno ing sel - e e en ial dynamics whe e obse a ion and e olu ion a e coupled.
Recen wo k has es ablished ha biological sys ems ou inely ope a e nea undamen al
measu emen limi s [17], wi h causal pa e ns exis ing below he Landaue h eshold whe e
hey esis bina y measu emen ye emain unc ionally decisi e h ough collec i e e ec s.
This sub-Landaue domain—popula ed by ephap ic coupling [18, 19], weak synap ic noise
[20], and quan um-like cohe ences [21, 22]—exhibi s s uc u ed nonde e minism ha classical
in o ma ion he modynamics canno adequa ely desc ibe.
He e we de elop a gene alized amewo k ea ing in o ma ion e asu e as dimensional
collapse in high-dimensional cohe ence ields. This concep can be unde s ood by analogy
o p incipal componen analysis in machine lea ning: jus as PCA p ojec s high-dimensional
da a on o lowe -dimensional p incipal componen s while p ese ing maximal a iance, mea-
su emen p ojec s high-dimensional s a es on o lowe -dimensional obse ables, wi h associ-
a ed in o ma ion loss and ene ge ic cos . Howe e , unlike PCA which is pu ely ma hema i-
cal, physical dimensional collapse is an i e e sible he modynamic p ocess go e ned by he
gene alized bounds we de i e. This e o mula ion:
•Ex ends Landaue ’s bound o con inuous phase-space mani olds
•Rein e p e s Maxwell’s demon as in e nal symme y b eaking
•P edic s sub-Landaue e asu e egimes h ough cohe ence edis ibu ion
•Uni ies measu emen , compu a ion, and he modynamics unde in o ma ion geome y
The amewo k connec s na u ally o ecen ad ances in unde s anding biological com-
pu a ion [23], whe e in elligence eme ges om main aining high-dimensional cohe ence in
3
egimes whe e empo al ine-s uc u e becomes undamen ally inaccessible o measu emen .
By ecognizing ha he demon canno emain ex e nal o he cohe ence ield i exploi s,
we esol e longs anding ensions be ween in o ma ion heo y and he modynamics while
e ealing new compu a ional possibili ies in he sub-Landaue domain.
2 Classical Fo mula ion and I s Limi a ions
2.1 S anda d Landaue Bound
In Landaue ’s s anda d o m [3, 15],
Ee ase ≥kBTln 2,(1)
e asu e is modeled as con ac ion om wo equip obable mic os a es o one. The unde lying
assump ions a e:
•Disc e e s a e space: In o ma ion exis s as well-de ined, sepa able bi s.
•Low-dimensional phase space: No hidden co ela ions o coupled deg ees o ee-
dom.
•The mal equilib ium: A uni o m ba h a empe a u e T.
•De e minis ic e asu e: One- o-one mapping om ini ial mic os a es o inal s a e.
The de i a ion p oceeds h ough he Szila d engine [16]: Conside a single-molecule ideal
gas con ined o a box o olume Va empe a u e T.
S ep 1 - Measu emen : Inse a pa i ion a he midpoin , de e mining which hal
con ains he molecule (one bi o in o ma ion). This s ep can be e e sible in p inciple.
S ep 2 - Ex ac ion: A ach a weigh o he pa i ion and allow iso he mal expansion.
The molecule pushes he pa i ion, pe o ming wo k W=kBTln 2 as i expands om V/2
o V.
4
S ep 3 - E asu e: To ope a e cyclically, he demon mus e ase i s memo y o which
side he molecule occupied. Remo ing he pa i ion and allowing he molecule o equilib a e
o e he ull olume inc eases en opy by ∆S=kBln 2, equi ing minimum hea dissipa ion
Q≥kBTln 2 o he en i onmen .
The ne wo k ex ac able is Wne =kBTln 2 −kBTln 2 = 0, es o ing he modynamic
consis ency. The engine con e s he mal ene gy o wo k only by u ilizing p e-exis ing in-
o ma ion; he e asu e s ep ensu es no ne iola ion o he second law.
2.2 Limi a ions o Biological Sys ems
These assump ions ail o sys ems go e ned by con inuous, high-dimensional cohe ence
dynamics—neu onal ields, biochemical eac ion ne wo ks, o op ical cohe ence sys ems—
whe e in o ma ion is encoded as phase alignmen , no symbol coun [17, 23].
Conside neu al popula ion coding. A ne wo k o Ncoupled oscilla o s can ep esen
s a es h ough hei collec i e phase con igu a ion ϕ= (ϕ1, . . . , ϕN). The ”in o ma ion” is
no disc e e bi s bu con inuous phase ela ionships encoding senso y inpu s, mo o plans,
o in e nal models. Measu ing his s a e o ex ac a bina y decision equi es p ojec ing he
high-dimensional mani old on o a low-dimensional obse able, des oying he phase ela ion-
ships ha cons i u e he ep esen a ion [24, 25].
Simila ly, p o ein olding na iga es ugged ene gy landscapes wi h as onomical num-
be s o con o ma ional s a es [26]. The olding ajec o y encodes in o ma ion abou he
na i e s a e h ough ansien in e media es and pa allel pa hways, bu measu ing he pa h
collapses i o a bina y ou come ( olded/un olded), e asing he dimensional s uc u e ha
enables e icien explo a ion.
In pho osyn he ic ene gy ans e , quan um cohe ence enables nea -uni y e iciency by
allowing exci ons o simul aneously sample mul iple pa hways [21, 27]. The cohe en supe -
posi ion exis s a ene gies ∼10−22 J, well below he Landaue limi . Measu emen su icien
o de e mine he speci ic pa hway des oys he cohe ence and educes e iciency—ye he
5
cohe ence causally de e mines he ou come.
2.3 The Ex e nal Demon P oblem
S anda d Maxwell’s demon o mula ions ea he demon as ex e nal o he sys em—a sep-
a a e agen wi h i s own memo y, pe o ming measu emen s ha lea e he sys em unpe -
u bed while ex ac ing in o ma ion. This sepa a ion becomes un enable in high-dimensional
cohe ence sys ems whe e:
1. The demon’s ”memo y” mus be encoded in he same physical subs a e as he sys em.
2. Measu emen is no passi e obse a ion bu ac i e p ojec ion ha al e s sys em dy-
namics.
3. The demon-sys em bounda y is a bi a y; bo h a e subsys ems o a la ge cohe ence
ield.
4. Feedback be ween demon and sys em c ea es sel - e e en ial dynamics whe e he dis-
inc ion dissol es.
As we will show, he demon is mo e accu a ely unde s ood as an in e nal a ac o —a
low-en opy submani old o he sys em’s phase space ha selec i ely ampli ies co ela ions.
The demon doesn’ obse e om ou side; i eme ges om wi hin as spon aneous symme y
b eaking in he cohe ence ield i sel .
3 High-Dimensional Dynamics and Cohe ence Fields
3.1 No a ion and Assump ions
Be o e de eloping he gene alized amewo k, we es ablish key de ini ions and ope a ional
assump ions:
6
Coa se-g ained phase-space olume: We wo k wi h a coa se-g ained desc ip ion a
esolu ion ∆xon he slow mani old o collec i e modes. The olume V(De ) ep esen s
he numbe o dis inguishable con igu a ions, no he ine-g ained Liou ille measu e. Di-
mensional collapse is a many- o-one logical map ealized h ough coupling o a ba h; he
unde lying Hamil onian mic oe olu ion p ese es ine-g ained olume pe Liou ille’s heo-
em.
Geome ic olume scaling: Fo e ec i e dimensionali y De and esolu ion ∆x, we
assume he accessible olume scales as a p oduc measu e o e e ec i e modes: V∼
(∆L/∆x)De whe e ∆Lis he ypical ampli ude scale. This holds o weakly coupled modes
o e ec i e ellipsoidal geome y in p incipal-componen coo dina es. Real biological mani-
olds may exhibi mo e complex scaling (e.g., ac al dimensions), bu he p oduc measu e
p o ides a conse a i e lowe bound on dimensional en opy; depa u es s eng hen a he
han weaken ou bounds.
E ec i e dimensionali y: We de ine De ia he pa icipa ion a io o he co ela ion
eigenspec um:
De =1
Pip2
i
(2)
whe e pia e no malized eigen alues o he phase co ela ion ma ix. A ixed esolu ion ∆x,
he coa se-g ained olume scales as V∝(∆x)−De (assuming a p oduc measu e o e ec i e
ellipsoidal geome y), jus i ying ln(Vp e/Vpos )∝De −D′.
E ec i e empe a u e: We de ine Te ope a ionally ia a luc ua ion-dissipa ion ela-
ion on he collec i e mode a equency ω0:
kBTe ≡Sη(ω0)
2γe
(3)
whe e Sηis he noise powe spec al densi y and γe is he co esponding dissipa i e esponse.
In he mal equilib ium, Te =T. Fo d i en sys ems wi h s uc u ed noise, Te can de ia e
om he ba h empe a u e.
7
ch onized con igu a ion whe e phases lock:
ϕdemon( )≈ [ϕsys em( )] (21)
o some unc ional ela ionship . The demon’s deg ees o eedom become en ained o
he sys em’s, educing he o al e ec i e dimensionali y:
Dco ela ed
e < Dsys em +Ddemon (22)
This dimensional educ ion IS he in o ma ion acquisi ion. The e is no sepa a e ”memo y”—
he memo y is he cons ained phase-space olume.
5.3 E asu e as Decohe ence
E asu e co esponds o b eaking he co ela ions—allowing he demon’s subsys em o deco-
he e and egain independence:
I(Xsys em;Xdemon)→0 (23)
In dynamical e ms, his means dis up ing he phase-locking ela ionship, allowing he
subsys ems o explo e hei ull indi idual phase spaces:
Dunco ela ed
e →Dsys em +Ddemon (24)
The dimensional expansion equi es ene gy injec ion (o en opy inc ease) acco ding o
he gene alized Landaue bound de i ed abo e. The demon canno escape his cos because
i is pa o he sys em, no ex e nal o i .
14
5.4 The Demon as A ac o
We can o malize he demon as a s able a ac o Ademon ⊂ M o al wi h basin o a ac ion
B. The sys em na u ally lows owa d his a ac o :
dx
d =−∇E(x) + η( ) (25)
whe e he ene gy landscape E(x) has a minimum a Ademon.
The demon’s ”ac ion” is hen spon aneous symme y b eaking— he sys em’s na u al
endency o minimize ene gy by o ming cohe ence s uc u es. No ex e nal agen is needed;
he demon eme ges om he ield’s in e nal dynamics.
In o ma ion ex ac ion occu s when he sys em eaches he a ac o :
x( → ∞)∈ Ademon (26)
and e asu e occu s when noise o ex e nal pe u ba ion kicks he sys em ou o he basin:
x( )/∈ B =⇒loss o co ela ion (27)
The he modynamic cos is he ene gy equi ed o des abilize he a ac o o e ill i s
basin—exac ly wha he gene alized Landaue bound p edic s.
6 Appa en Sub-Landaue unde Dimensional Redis-
ibu ion
6.1 Cohe ence Redis ibu ion
In high-dimensional sys ems, appa en iola ions o he Landaue bound can occu locally
when cohe ence is edis ibu ed a he han des oyed. Conside a sys em wi h mul iple
15
equency modes:
Φ(x, ) = X
k
Akei(k·x−ωk +ϕk)(28)
”E asu e” in one mode can co espond o ans e o ano he mode a he han dissipa-
ion:
Ak→0, Ak′→Ak′+ ∆A(29)
The o al cohe ence (in eg a ed ac oss modes) emains cons an :
X
k|Ak|2= cons (30)
This is dimensional conse a ion: deg ees o eedom a e no elimina ed bu econ igu ed.
6.2 Sub-Landaue Appa en E asu e
When measu ed locally a mode o subsys em α, he appa en e asu e ene gy can all below
kBTln 2 because he los in o ma ion is no he malized bu ans e ed elsewhe e in he
ield:
E(α)
e ase =kBT(α)
e ln Vp e
α
Vpos
α< kBTln 2 (31)
C ucially, his is an appa en local iola ion only. The o al ene gy dissipa ed (and
ac ual he modynamic cos ) includes bo h he local collapse and he compensa ing expansion
elsewhe e. Fo each subsys em unde going collapse, he en opy change is nega i e:
∆Sα=kBln Vpos
α
Vp e
α<0 (32)
Globally ac oss all modes and subsys ems, including he en i onmen , en opy is con-
se ed o inc eased:
X
α
∆Sα+ ∆Sen ≥0 (33)
The appa en sub-Landaue collapse in one subspace mus be o se by an i-collapse
16
(expansion) in o he subspaces o hea dissipa ion o he en i onmen . Rew i ing in e ms
o ene gy cos s (whe e posi i e e ms e lec dissipa ion compensa ing collapse):
X
α
kBT(α)
e ln Vp e
α
Vpos
α≥0 (34)
whe e he sum uns o e all subsys ems α. This sa is ies he second law globally while
pe mi ing local appa en iola ions.
Reconcilia ion wi h in o ma ion he modynamics: This amewo k is ully com-
pa ible wi h he Sagawa-Ueda gene alized second law o sys ems wi h eedback [11, 12].
Thei wo k shows ha in o ma ion gain I educes he minimum wo k equi ed:
⟨W⟩ ≥ ∆F−kBTI (35)
In ou amewo k, cohe ence o ma ion IS in o ma ion gain, whe e he coa se-g ained mu ual
in o ma ion sa is ies I≈ln Vco /Vunco ≈ln V(De )/V (D′), and dimensional edis ibu ion
IS he mechanism by which his in o ma ion educes local wo k. The key di e ence: a he
han an ex e nal eedback con olle , he ”demon” is an in e nal a ac o ha spon aneously
o ms co ela ions. The dimensional accoun ing makes explici wha Sagawa-Ueda ea as
mu ual in o ma ion—bo h amewo ks espec he global second law while enabling local
appa en sub-Landaue cos s h ough co ela ion s uc u e.
This explains how biological sys ems achie e appa en sub-Landaue e asu e [17]: hey
ope a e in dimensional conse a ion mode, ecycling co ela ion s uc u e ac oss coupled
oscilla o s a he han dissipa ing i o hea . The los deg ees o eedom in one subsys em
a e econ igu ed in o ano he , no des oyed. The ac ual o al cos espec s Landaue ; he
appa en local cos can all below i .
17
6.3 S uc u ed Nonde e minism
When he unde lying dynamics exhibi s uc u ed nonde e minism—phase-locked noise,
s ochas ic esonance [31], o quan um-like cohe ences— he e ec i e e asu e empe a u e Te
can di e om he physical empe a u e T.
Fo sys ems d i en by co ela ed noise:
⟨ηi( )ηj( ′)⟩= Γijδ( − ′) (36)
wi h co ela ion ma ix Γij, he e ec i e empe a u e scales wi h he noise s eng h mod-
ula ed by he dissipa ion:
kBTe ∝λmax(Γ)
γe
(37)
whe e λmax is he la ges eigen alue o he noise co ela ion and γe is he e ec i e damp-
ing on he collec i e mode. When noise is an i-co ela ed o shows empo al s uc u e, his
can yield Te < T, enabling sub- he mal appa en e asu e a he local le el.
This connec s o he sub-Landaue amewo k [17, 23]: pa e ns exis p ecisely because
hey exploi s uc u ed noise below he he mal loo , main aining cohe ence h ough s ochas-
ic esonance a he han de e minis ic con ol.
6.4 Conc e e Example: Coupled Bis able Oscilla o s
To illus a e he mechanism conc e ely wi h explici global en opy accoun ing, conside a
oy model:
Toy model: Two o e damped bis able oscilla o s (x, y) wi h coupled po en ial:
U(x, y;λ) = U0(x;λ)+U0(y; 0) + K
2(x−y)2
18
whe e U0(z;λ) = 1
4z4−z2−λz is a double-well po en ial, and he dynamics a e:
˙x=−∂U
∂x +ηx( )
˙y=−∂U
∂y +ηy( )
wi h independen Gaussian noise ⟨ηi( )ηj( ′)⟩= 2Dδijδ( − ′).
Fo s ong coupling K≫1, he slow mani old sa is ies x≈ywi h e ec i e
dimensionali y De ≈1 (one collec i e mode). De ine cohe ence as he co ela ion
coe icien ρ= co (x, y)/p a (x) a (y)∈[0,1]; as Kinc eases, ρ→1.
E asu e p o ocol: P ojec xon o sign(x) (le s igh well).
Local en opy: Fo uncoupled oscilla o s (K= 0), each has wo wells o equal
olume, so e asu e cos s:
∆Suncoupled
x=−kBln 2 =⇒Ee ase =kBTln 2
Wi h coupling (K≫1), he cons ain x≈y educes he accessible olume
p e-e asu e. The e ec i e olume con ac ion becomes:
ln Vp e
x
Vpos
x≈ln(2ρ) wi h ρ<1
gi ing appa en local cos :
Eappa en
e ase =kBTln(2ρ)< kBTln 2
Global accoun ing: The ”missing” en opy kBln(1/ρ) appea s in wo places:
1. T ans e se mode expansion: Collapsing xc ea es luc ua ions in he
an i-phase mode (x−y). The cons ain elaxa ion allows ∆S⊥≈kBln(1/ )>
19
0.
2. Hea o ba h: The coupling o ce −K(x−y) does wo k du ing collapse,
dissipa ing ene gy ∼K⟨(x−y)2⟩ o he he mal ba h, con ibu ing en opy
∆Sba h ∼kB(Ewo k/T)>0.
The global en opy balance sa is ies:
∆S o al = ∆Sx+ ∆S⊥+ ∆Sba h =−kBln 2 + kBln(1/ )+kBln(1/ )≥0
o ≤1/√2, wi h equali y app oached in he quasis a ic limi .
This exempli ies dimensional edis ibu ion: cohe ence educes he appa en local cos
(ac ual in o ma ion emo ed is less han one ull bi due o cons ain s), while expansion
elsewhe e es o es global he modynamic consis ency. The local obse e sees sub-Landaue
e asu e; he global accoun ing espec s he second law. Appendix A p o ides nume ical
alida ion ia Lange in simula ion.
6.5 Obse able Regimes
We can dis inguish h ee egimes:
1. Supe -Landaue (Ee ase ≫kBTln 2): High-dimensional collapse, i e e sible he -
maliza ion, classical e asu e.
2. Nea -Landaue (Ee ase ∼kBTln 2): Bi -le el ope a ions, s anda d compu ing, equi-
lib ium he modynamics applies.
3. Sub-Landaue (Ee ase < kBTln 2): Cohe ence edis ibu ion, dimensional conse a-
ion, s uc u ed nonde e minism. Local e asu e enabled by global conse a ion.
Biological sys ems ope a e p ima ily in he sub-Landaue egime [17], exploi ing cohe -
ence edis ibu ion o e icien compu a ion while paying ull Landaue cos only a inal
20
measu emen collapse [23].
7 Implica ions and Obse able P edic ions
7.1 Biological In o ma ion P ocessing
This amewo k explains se e al puzzling ea u es o biological compu a ion:
Neu al e iciency: The b ain ope a es a ∼20 W while pe o ming compu a ions ha
would equi e megawa s in silicon [33]. Sub-Landaue e asu e h ough cohe ence edis ibu-
ion explains his e iciency: neu al popula ions main ain oscilla o y cohe ence ha encodes
in o ma ion wi hou con inuous w i e-e ase cycles [34, 35], collapsing only a decision poin s.
Wo king memo y, o ins ance, ope a es h ough sus ained oscilla o y ac i i y pa e ns ha
ep esen in o ma ion in high-dimensional neu al s a e spaces wi hou equi ing disc e e s o -
age ope a ions a each momen .
Analog ad an age: Biological sys ems a o analog o e digi al compu a ion. This
makes sense: analog cohe ence dynamics na u ally ope a e in he sub-Landaue egime,
while digi al equi es disc e e w i es a ull Landaue cos .
Collec i e phenomena: Ephap ic coupling [18], weak synap ic noise [20], and pop-
ula ion cohe ence all ope a e below indi idual de ec ion h esholds ye causally in luence
ou comes. These a e sub-Landaue pa e ns edis ibu ed ac oss popula ions.
7.2 Quan um Biology
The amewo k p o ides a he modynamic ounda ion o quan um e ec s in biology. Quan-
um cohe ence in pho osyn hesis [21, 27] exis s a ∼10−22 J, o de -o -magni ude below he
Landaue limi . While he ole o quan um cohe ence in biological unc ion emains ac i ely
deba ed [22]—wi h c i ics a guing he mal decohe ence should domina e a physiological
empe a u es— ecen expe imen al and heo e ical ad ances con inue o demons a e cohe -
ence e ec s pe sis ing longe han classical models p edic [27]. Eme ging wo k sugges s
21
quan um e ec s may pe sis h ough en i onmen al s uc u ing and p o ec i e mechanisms
[36, 37], whe e he biological milieu i sel modula es decohe ence imescales o enable unc-
ional quan um dynamics.
In ou amewo k, quan um cohe ence is main ained p ecisely because i ope a es below
he measu emen h eshold—obse a ion would injec ene gy ha des oys he cohe ence.
The exci on explo es mul iple pa hways simul aneously h ough dimensional supe posi ion,
collapsing only a he eac ion cen e . The ull Landaue cos is paid only a he inal cha ge
sepa a ion, no du ing cohe ence p opaga ion. Whe he he mal decohe ence domina es
depends on he a io o decohe ence ime o unc ional imescale; in op imized biological
sys ems, he la e can be su icien ly sho o exploi cohe ence dynamics [22], wi h e-
cen e idence sugges ing quan um e ec s ex end beyond pho osyn hesis o ol ac ion, enzyme
ca alysis, and po en ially neu al p ocesses [38]. The ole o quan um-like cohe ence s a es
in biological o ganiza ion and in o ma ion p ocessing ep esen s a undamen al challenge o
classical educ ionis app oaches [39].
7.3 Expe imen al P edic ions
The amewo k makes es able p edic ions:
P edic ion 1 - Dimensional Scaling: E asu e ene gy should scale wi h los dimen-
sionali y, no bi coun . Sys ems wi h highe De equi e p opo ionally mo e ene gy o
collapse o he same D′.
Tes able in a i icial pho onic sys ems o coupled oscilla o ne wo ks by measu ing hea
dissipa ion du ing dimensional p ojec ion.
P edic ion 2 - Cohe ence Dependence: E asu e cos should dec ease wi h inc easing
cohe ence ρ. Highly synch onized sys ems ha e lowe e ec i e dimensionali y, educing he
dimensional gap ha mus be b idged du ing collapse.
Tes able in neu al popula ions by co ela ing spike- ime cohe ence wi h me abolic ma k-
e s ( ia calcium imaging o PET).
22
P edic ion 3 - Sub-Landaue De ec ion: Neu al popula ions should de ec signals
wi h Esignal ≪kBTln 2 h ough empo al in eg a ion. De ec ion h esholds all below Lan-
daue limi when empo al in eg a ion windows exceed ∼100ms ( es able ia psychophysical
expe imen s wi h con olled noise). S ochas ic esonance in neu al sys ems p o ides a mech-
anism o such de ec ion [32, 20]. Recen p oposals o quan um e ec s in neu al p ocessing
[36] sugges addi ional mechanisms o sub- h eshold signal de ec ion, hough expe imen al
alida ion emains an ac i e esea ch on ie .
P edic ion 4 - Collapse Signa u es: Sudden dimensional educ ion e en s should be
de ec able in neu al eco dings, co ela ed wi h decision/ou pu e en s. These should show
cha ac e is ic ene gy dissipa ion ma ching Landaue cos o bi s w i en.
P edic ion 5 - Me abolic E iciency: B ain egions wi h highe De should show
be e ope a ions-pe -wa e iciency (measu able ia combined MRI/PET imaging).
7.4 Hie a chical Resonance and Biological Implemen a ion
Biological sys ems implemen cohe ence compu a ion h ough nes ed oscilla o y hie a chies
[40]. Neu al popula ion oscilla ions span del a (0.5-4 Hz), he a (4-8 Hz), alpha (8-13 Hz),
be a (13-30 Hz), and gamma (30-100 Hz) bands, wi h high- equency oscilla ions ex ending
o 500 Hz in specialized con ex s. These hy hms o ganize h ough c oss- equency coupling,
whe e slow oscilla ions modula e he ampli ude and phase o as e hy hms.
C oss- equency coupling binds in o ma ion ac oss imescales wi hou equi ing measu e-
men a indi idual le els. The compu a ional capaci y scales wi h he p oduc o pa icipa -
ing equency bands and hei e ec i e dimensionali ies:
D o al ≈
Nle els
Y
n=1
D(n)
e (38)
This a chi ec u e enables exponen ial compu a ional capaci y h ough empo al mul i-
plexing a he han spa ial scaling, all main ained in he unmeasu able egime un il s a egic
23
4. Dimensional conse a ion: High-dimensional sys ems ope a e in a egime whe e
in o ma ion can be ans e ed ac oss modes wi hou he maliza ion, enabling appa en
sub-Landaue e asu e locally while espec ing he modynamics globally.
This amewo k uni ies Maxwell’s demon, Landaue ’s p inciple, and cohe ence he mody-
namics unde in o ma ion geome y. I explains biological compu a ional e iciency, p o ides
he modynamic ounda ions o quan um biology, and sugges s new pa adigms o compu ing
based on dimensional conse a ion a he han bi manipula ion.
The deep insigh : bi s a e p ojec ions, no p imi i es. In o ma ion exis s undamen-
ally as high-dimensional cohe ence s uc u e. Disc e eness eme ges h ough measu emen
collapse, which is i sel a he modynamic p ocess go e ned by dimensional en opy loss.
Maxwell’s demon exploi s his s uc u e no by iola ing he modynamics bu by ope a ing
wi hin i as an in e nal symme y b eake , insepa able om he ield i appea s o con ol.
The u u e o in o ma ion he modynamics lies in unde s anding dynamics on cohe ence
mani olds—how dimensional s uc u es o m, e ol e, collapse, and edis ibu e. This is he
domain whe e in o ma ion, ene gy, and en opy uni e, whe e classical and quan um become
pe spec i es on he same geome ic eali y, and whe e he appa en pa adoxes o Maxwell’s
demon dissol e in o he na u al low o a high-dimensional uni e se owa d i s a ac o s.
Acknowledgmen s
The au ho hanks he e iewe s o cons uc i e eedback ha signi ican ly imp o ed his
manusc ip .
Funding
This esea ch did no ecei e any speci ic g an om unding agencies in he public, com-
me cial, o no - o -p o i sec o s.
30
Decla a ion o compe ing in e es
The au ho decla es no compe ing inancial in e es s o pe sonal ela ionships ha could
ha e in luenced his wo k.
Decla a ion o gene a i e AI use
Du ing p epa a ion he au ho used Claude (An h opic) o li e a u e e iew, ma hema ical
o mula ion, and edi ing. Cha GPT (OpenAI) and G ok (xAI) we e used o independen
e i ica ion, e o checking, and c i ical e iew o he manusc ip . All con en was e iewed
and alida ed by he au ho , who akes ull esponsibili y o he published a icle.
Da a a ailabili y
The nume ical simula ion code o Appendix A is p o ided in ull wi hin he manusc ip and
cons i u es he comple e implemen a ion. The code can be ep oduced independen ly using
he speci ied pa ame e s and any s anda d Py hon en i onmen wi h NumPy. No addi ional
da a o analysis sc ip s beyond hose p esen ed in he manusc ip we e used in his s udy.
P ep in a ailable a : h ps://doi.o g/10.5281/zenodo.17309152
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A Nume ical Valida ion: Coupled Bis able E asu e
This appendix p o ides quan i a i e alida ion o appa en sub-Landaue e asu e h ough
nume ical simula ion o he coupled bis able oy model p esen ed in Sec ion 6.3. We imple-
men he ull s ochas ic dynamics ia Lange in equa ions and demons a e ha local hea
dissipa ion alls below kBTln 2 while global en opy p oduc ion sa is ies he second law.
A.1 Sys em Se up
Conside wo o e damped deg ees o eedom x( ), y( ) e ol ing in coupled double-well po-
en ials:
U(x, y;λ) = U0(x;λ)+U0(y; 0) + K
2(x−y)2(43)
36
whe e he single-well po en ial is:
U0(z;λ) = 1
4z4−z2−λz (44)
The con ol pa ame e λ( ) ac s only on x, implemen ing he e asu e p o ocol. The
coupling s eng h Kcon ols cohe ence: s ong coupling (K≫1) o ces x≈y, educing
e ec i e dimensionali y.
The o e damped Lange in dynamics a e:
γ˙x=−∂U
∂x +p2γkBTxξx( ) (45)
γ˙y=−∂U
∂y +p2γkBTyξy( ) (46)
whe e ξx,y a e independen Gaussian whi e noises wi h ⟨ξi( )ξj( ′)⟩=δijδ( − ′). We se
Tx=Ty=T o he base case; he e ec i e empe a u e Te on he slow in-phase mode
eme ges ia he luc ua ion-dissipa ion ela ion and equals Ta equilib ium.
A.2 E asu e P o ocol
The p o ocol consis s o ou phases:
1. Equilib a ion: S a a λ= 0 o ∈[−τeq,0] o each he mal equilib ium in he
symme ic double well.
2. Bias amp (e asu e):λ( ) = λmaxs( /τ) o ∈[0, τ] wi h smoo h amp unc ion
s(u) = 3u2−2u3∈[0,1]. This biases he xwell o a o x > 0, implemen ing e asu e
o he posi i e s a e.
3. Hold: Main ain λ=λmax o ime τhold o allow ull elaxa ion in o x>0.
4. Unbias: (Op ional) Re u n o λ= 0 o comple e a ull logical cycle.
37
In he uncoupled case (K= 0) and quasi-s a ic limi , he minimal a e age hea eleased
o he ba h sa is ies ⟨Qx⟩≥kBTln 2 (s anda d Landaue ). Wi h coupling K > 0, we p edic :
⟨Qx⟩< kBTln 2 (appa en sub-Landaue ) (47)
⟨Qx⟩+⟨Qy⟩ ≥ kBTln 2 (global compliance) (48)
A.3 Hea and Wo k Bookkeeping
Following Sekimo o’s s ochas ic ene ge ics o malism, we decompose ene gy changes along
ajec o ies. Fo coo dina e i∈ {x, y}wi h S a ono ich calculus:
δQi=∂U
∂i ◦di (49)
whe e ◦deno es he S a ono ich p oduc (midpoin ule in disc e e in eg a ion). We ollow
he con en ion whe e Qi ep esen s hea abso bed by subsys em i; he hea dissipa ed o
he ba h is −Qi. In ou igu es and ables, we epo he dissipa ed hea (posi i e alues
indica e ene gy lea ing he subsys em o he en i onmen ).
The wo k pe o med by he ex e nal con ol is:
δW =∂U
∂λ ˙
λ d =−x˙
λ d (50)
Ene gy conse a ion equi es ∆U=W−(Qx+Qy) along each ajec o y. We es ima e
ensemble a e ages ⟨Qx,y⟩by ime in eg a ion o e many independen ajec o ies.
A.4 Nume ical Pa ame e s
We wo k in dimensionless uni s wi h γ= 1. A obus pa ame e egime demons a ing he
e ec :
•Double-well shape: a= 1, b= 2 (minima nea ±1)
38
•Coupling s eng hs: K∈ {0,0.5,1.0,2.0}
•Tempe a u e: T= 0.25 (mode a e he mal luc ua ions)
•Maximum bias: λmax = 0.6
•Ramp du a ion: τ= 5000 imes eps wi h d = 10−3
•Hold ime: τhold = 2000 s eps
•Equilib a ion: τeq = 5000 s eps
•Ensemble size: N aj = 2000 ajec o ies
Longe amp imes τapp oach he quasi-s a ic lowe bounds mo e closely. On a s an-
da d lap op CPU, N aj = 500 ajec o ies akes app oxima ely 5-10 minu es and p o ides
adequa e s a is ical alida ion; he ull N aj = 2000 ensemble equi es app oxima ely 30-40
minu es bu imp o es p ecision.
A.5 Obse able P edic ions
P ima y me ic: A e age local hea ⟨Qx⟩ s coupling K. Expec ed beha io :
•K= 0: ⟨Qx⟩ ≈ kBTln 2 ≈0.173 (s anda d Landaue )
•K > 0: ⟨Qx⟩<0.173 (appa en sub-Landaue )
•All K:⟨Qx+Qy⟩≳0.173 (global compliance)
Dimensional educ ion indica o : Pa icipa ion a io om co a iance eigen alues
λ1,2o (x, y) du ing he amp:
De =(λ1+λ2)2
λ2
1+λ2
2
(51)
As Kinc eases, De →1 (one e ec i e collec i e mode).
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The global dissipa ion obeys ou gene alized geome ic bound om Eq. (*):
⟨Qx+Qy⟩ ≥ kBTe ln V(De )
V(D′)(53)
eco e ing s anda d Landaue when K→0 and De →2 ( wo independen bi s collapsing
o one). This nume ical alida ion demons a es ha dimensional edis ibu ion enables
appa en sub-Landaue e asu e locally while p ese ing global he modynamic consis ency.
46
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maxwell cohe ence demon. ex
Decla a ion o Compe ing In e es s
Manusc ip Ti le: Maxwell's Demon in a High-Dimensional Uni e se: Cohe ence Collapse
and he Limi s o Landaue E asu e
The au ho decla es ha he has no known compe ing inancial in e es s o pe sonal
ela ionships ha could ha e appea ed o in luence he wo k epo ed in his pape .
Au ho : Ian Todd
Da e: Oc obe 10, 2025
Decla a ion o In e es S a emen
