Distinguishable Cash Bosonic Bitcoin and Fermionic Non-fungible Token Zae Young Kim

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Distinguishable Cash, Bosonic Bitcoin, and Fermionic Non-fungible Token

Zae Young Kim

Center for Quantum Spacetime, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul 04107, Korea

Jeong-Hyuck Park∗

Department of Physics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul 04107, Korea

Modern technology has brought novel types of wealth. In contrast to hard cash, digital currency

does not have a physical form. It exists in electronic forms only. To date, it has not been clear

what impacts its ongoing growth will have, if any, on wealth distribution. Here, we propose to

identify all forms of contemporary wealth into two classes: distinguishable or identical. Traditional

tangible moneys are all distinguishable. Financial assets and cryptocurrencies, such as bank deposits

and Bitcoin, are boson-like, while non-fungible tokens are fermion-like. We derive their ownership-

based distributions in a uniﬁed manner. Each class follows essentially the Poisson or the geometric

distribution. We contrast their distinct features such as Gini coeﬃcients. Furthermore, aggregating

diﬀerent kinds of wealth corresponds to a weighted convolution where the number of banks matters

and Bitcoin follows Bose–Einstein distribution. Our proposal opens a new avenue to understand

the deepened inequality in modern economy, which is based on the statistical physics property of

wealth rather than the individual ability of owners. We call for veriﬁcations with real data.

Introduction.—When two one-dollar banknotes are

randomly gifted to two people, there occur total four

possible ways of distributions. While counting so, it has

been naturally assumed that both notes are distinguish-

able from each other, since they are for sure distinct phys-

ical objects, not to mention the diﬀerent serial numbers

printed on them. In contrast, when two cents are cred-

ited to a pair of savings bank accounts, there are three

possibilities, because the two cents as deposits are in-

distinguishable. Deposits do not have a physical form.

They exist in the form of abstract numbers by ‘claim’

and ‘trust’ between the bank and the account holders.

While one’s can add up to a natural number, say k∈N,

1 + 1 + ··· + 1 = k , (1)

all the one’s are intrinsically identical and indistinguish-

able from one another. The notion of being indistin-

guishable, or interchangeably identical, is a fundamental

property of elementary particles in physics: bosons can

share quantum states but fermions subject to the Pauli

exclusion principle cannot. Consequently, their statisti-

cal distributions diﬀer signiﬁcantly. While the identical

property holds certainly for particles at quantum scale,

there appears no clear-cut limit of applicability to larger

macroscopic objects.

In this paper, we propose to identify all kinds of wealth

into two classes: distinguishable or identical. All the tra-

ditional tangible moneys i.e. hard cash including minted

coins and banknotes are of physical existence and belong

to the distinguishable class. In contrast, ﬁnancial assets

like bank deposits, stocks, bonds, and loans belong to the

boson-like identical class. Furthermore, all the electronic

forms of wealth share the identical property. At deep

down level of information technology or atomic physics,

∗park@sogang.ac.kr

they comprise of chain of bits which have ﬁnite length.

The pieces of information stored are accordingly limited

mostly to the amounts and, hence, are abstract like the

deposit or the natural number (1). With no restriction

on the amount of possession, cryptocurrencies, e.g. Bit-

coin [1] are boson-like. Contrarily, having unique digital

identiﬁers, non-fungible tokens (NFTs) may be identiﬁed

as fermions. Having said so, we shall demonstrate that

generic identical wealth can be universally and eﬀectively

described by Gentile statistics [2] which postulates a cut-

oﬀ for the maximal amount of possession.

It is an established fact that distinguishable, bosonic,

and fermionic particles follow respectively the Maxwell–

Boltzmann, Bose–Einstein, and Fermi–Dirac statistics,

which are all about the number of the particles them-

selves for a given energy. On the contrary, our primary

interest in this work is to derive the ownership-based dis-

tributions of wealth, i.e. the number of owners who pos-

sess a certain amount of wealth, while the owners are

assumed to be always distinguishable. Further, it is our

working assumption that wealth is distributed in a ‘ran-

dom’ manner. This should be the case if ideally the own-

ers were all equal. It goes beyond the scope of the present

paper to test the hypothesis against real data.

Basic scheme through elemental examples.—We start

with an elementary example of distributing Mnumber of

minted one-cent coins to Nnumber of people in a random

manner. We let nkbe the number of people each of whom

owns knumber of coins, k= 0,1,2,···. As we focus on

‘private ownership’ meaning no allowance of sharing, the

opposite notion “kn” does not make sense (except kn=1),

which in a way breaks the symmetry between people and

coins both of which are distinguishable. There are two

constraints nk’s satisfy

P∞

k=0 nk=N , P∞

k=0 knk=M . (2)

arXiv:2211.00291v2 [econ.GN] 9 Feb 2023

Irrespective of our notation, an eﬀective upper bound

in the sums exists such as 0 ≤k≤M. Our primary

aim is to compute the total number of all possible or

‘degenerate’ ways of distributions for a given set {nk’s}.

Hereafter, generically for any kinds of wealth, we denote

such a total number by Ω and further factorise it into two

numbers, Ω = Υ ×Φ, where Υ is all about the group-

ing of the owners into {nk’s}and thus is independent of

the sorts of wealth. The properties of wealth are to be

reﬂected in Φ. Speciﬁcally, the total number of possi-

ble cases for the Nnumber of people to be grouped into

n0, n1, n2,··· is

Υ = N!

n0!n1!n2!··· =N!

Q∞

k=0 nk!.(3)

While so, that for the Mcoins to be grouped into

1,1,··· ,1

| {z }

,2,2,··· ,2

| {z }

,······ k, k, ··· , k

| {z }

,··· ,(4)

is, as the coins are distinguishable,

Φ = M!

(1!)n1(2!)n2··· =M!

Q∞

k=1(k!)nk.(5)

Crucially, for each case in Υ, any of Φ can equally oc-

cur. Thus, the total number of possible distributions for

a given set {nk’s}is the product ΥΦ = Ω. The degen-

eracy Φ as counted in (5) is signiﬁcant since it depends

on nk’s. Insigniﬁcant degeneracies that are independent

of nk’s may be taken into account which will multiply

Φ by an overall constant. For example, extra distinc-

tions depending on whether the distribution of each coin

occurs in the morning or afternoon will give an overall

factor 2Mto Φ. Yet, our primary interest is to obtain the

most probable distribution of nk. Following the standard

analysis in statistical physics at equilibrium, e.g. [3], we

shall assume Nto be suﬃciently large, apply the varia-

tional method induced by δnkto ln Ω = ln Υ + ln Φ, and

acquire the extremal solution. Accordingly, any insignif-

icant degeneracy independent of nk’s becomes irrelevant

and ignorable. It merely shifts ln Φ by a constant.

We turn to savings accounts. We consider the Mcents

to be now credited to distinguishable Nsavings accounts.

Since deposits are boson-like identical, the total number

of possible distributions Ω is essentially Υ (3) itself up to

multiplying an insigniﬁcant overall constant. This irrel-

evant degeneracy can arise when the bank accounts keep

records of all the details of the crediting of the deposits,

e.g. the time of transaction, which would make the cred-

ited Mcents to appear seemingly distinguishable. How-

ever, all the information of each credit are recorded in

a chain of bits which has a ﬁnite length, say l=l0+l1

that decomposes into l0for the very record of the amount

kand l1reserved for any extra information. While the

former is rigidly ﬁxed, the extra pieces of information

are rather stochastic and hence contribute to ln Φ by a

constant shift, l1ln 2, which is hence ignorable.1

Lastly, fermion-like wealth or NFTs set M= 1 and

thus ﬁx the ownership-based distribution rather trivially:

nk= (N−1)δ0

k+δ1

k. Below, for each kind of wealth

we shall introduce what we call the “Gentile” parameter,

Λ∈N, which sets an upper bound on the possession num-

ber kas 0 ≤k≤Λ and interpolates boson at Λ = ∞and

fermion at Λ = 1. For distinguishable traditional moneys

in a ‘free’ country, the parameter may be set to coincide

with the total number of each kind, e.g. Min (2), or

to be less by law. However, electronic forms of wealth

can transform to one another. For example, the total

amount of deposits at a bank is not ﬁxed due to the ex-

ternal transfers between accounts at diﬀerent banks. The

total amount of each Bitcoin UTXO (Unspent Transac-

tion Output) is not ﬁxed either, since they can “combine”

and “split” to other UTXOs [1]. Thus, the total number

of each species of identical wealth is not a constant. For

this reason and also a technical reason later to justify the

approximation of ln nk!'nkln(nk/e), we shall keep Λ

as an independent key parameter which characterises, as

a matter of principle, boson-like or fermion-like identical

wealth.

Master formula.—For a unifying general analysis, we

consider distinguishable and identical wealth together.

We call each unit of wealth an object and postulate that

there are D=d+¯

ddistinct kinds of objects: dof them

are distinguishable and ¯

dof them are identical. We label

them by a capital index, I= 1,2,··· , D, which decom-

pose into small ones, I= (i, d + ¯ı) where i= 1,2,··· , d

for the distinguishable species and ¯ı= 1,2,··· ,¯

dfor

the identical species. An I-th kind object has value

wI∈N. For example, the present-day euro coin series

set d= 8 with w1= 1, w2= 2, ··· , w8= 200 in the unit

of cent. We then denote a generic ownership over them

by a D-dimensional non-negative integer-valued vector,

k= (k1, k2,··· , kD) of which each component kIdenotes

the number of owned Ith-kind objects and is bounded by

a cutoﬀ Gentile parameter: 0 ≤kI≤ΛI. In particular,

we set ΛI=∞for bosonic Iand ΛI= 1 for fermionic

I. We let n~

kbe the number of the owners with such a

ownership ~

k. The total number of owners is then

N=X

k≡

Λ1

k1=0

Λ2

k2=0 ···

ΛD

kD=0

k,(6)

and the total number of the Ith-kind objects is

MI=X

kIn~

k≡NmI.(7)

1In this reason, we prefer to say credits are boson-like rather than

(precisely) bosons. Further, we note that the extra information

is generically postdictive: they do not preexist before the trans-

actions take place, or before the ownerships settle down.

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DistinguishableCash,BosonicBitcoin,andFermionicNon-fungibleTokenZaeYoungKimCenterforQuantumSpacetime,SogangUniversity,35Baekbeom-ro,Mapo-gu,Seoul04107,KoreaJeong-HyuckParkDepartmentofPhysics,SogangUniversity,35Baekbeom-ro,Mapo-gu,Seoul04107,KoreaModerntechnologyhasbroughtnoveltypesofwealth.Incontra...

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