# Low-energy Chern-Simons-Proca theory

Diego R. Granado<EMAIL_ADDRESS>Institute of Research and
Development, Duy Tan University, Da Nang 550000, Viet Nam  Faculty of Natural
Sciences, Duy Tan University, Da Nang 550000, Viet Nam  Everton M. C. Abreu
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal Rural do
Rio de Janeiro, 23890-971, Seropédica, RJ, Brazil  Departamento de Física,
Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil
Programa de Pós-Graduação Interdisciplinar em Física Aplicada, Instituto de
Física, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ,
Brazil  A. Yu. Petrov<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970, João Pessoa,
Paraíba, Brazil

###### Abstract

Some time ago, the infrared limit of the Abelian Chern-Simons-Proca theory was
investigated. In this letter, we show how the Chern-Simons-Proca theory can
emerge as an effective low energy theory. Our result is obtained by means of a
procedure that takes into account the proliferation, or dilution, of
topological defects presented in the system.

topological defects, Chern-Simons-Proca theory, Julia-Toulouse approach

###### pacs:

11.27.+d, 11.10.Lm, 03.50.Kk

##  I Introduction

In a very interesting paper, the infrared limit of the Abelian Chern-Simons-
Proca (CSP) theory was investigated in Niemi:1994nw . By starting with a
regular Abelian CS theory and adding a Proca mass term to the gauge field, the
consequence was the introduction of a new dimensional parameter which is an
important perturbation of the CS action at the infrared, or long-distance,
limit. As it was pointed in Niemi:1994nw , this procedure could have relevant
consequences in the quantum Hall effect or in high-temperature
superconductivity, where it could, for instance, parametrize finite size
effects in an experimental framework. Also in Niemi:1994nw , it was found that
in the infrared limit of the Abelian CSP theory, the topological quantum
mechanical models investigated in Dunne:1989hv  can be recovered.

In this letter, we have used the Julia-Toulouse (JT) mechanism to achieve an
action whose low energy degrees of freedom are determined by the Abelian CSP
theory. The idea behind the JT mechanism is the description of how the
dynamics of the topological defects presented in the system can influence it.
Superconductors can be seen as one example of this procedure. Proliferation of
vortices on the sample can destroy the superconductivity state. As it will be
formally presented in the next section, JT mechanism can be used to describe
this phase transition between a superconductivity state and Maxwell free
theory due to vortices proliferation. In order to achieve this result, an
activation term, which looks like $\eta^{\mu}{\cal O}\eta_{\mu}$, and
describing the vortices dynamics, is introduced. Here, $\eta$ is the field
responsible for introducing vortices in the system, so that the integration
over $\eta$ brings new terms to the gauge theory.

In CW ; CWours ; Braga:2016atp , this operator ${\cal O}$ was treated as a
derivative expansion. In these papers, the goal was to study the influence of
topological defects in the lower regime of the theory. As such, the authors
only considered the low energy fluctuations of $\eta$. As a consequence a
Thirring-like interaction term was generated. Here we also took into account
the low energy fluctuations but we considered a Chern-Simons (CS) interaction
term for the currents. From this point, we demonstrated that, as a result, we
have a theory whose lower energy regime is described by the Abelian CSP
theory.

This paper is organized as follows: in section II, we reviewed the work
presented in Braga:2016atp  about the condensation of topological defects in
regular superconductors. In section III, we show how the mechanism used to
describe superconductors can be used to obtain an electrodynamics whose low
energy regime is described by the Abelian CSP theory. Finally, in the section
IV, we present our comments and conclusions.

##  II Short review on the condensation of topological defects

In this section, we present a short review on how the JT mechanism describes a
superconductor phase transition111A detailed review can be found in
Braga:2016atp ; Granado:2019bqk . As well known, the proliferation of vortices
in a superconductor sample can destroy the superconductivity state. Hence, the
system goes from a superconductivity state to a Maxwell free system. In our
approach, the proliferation of vortices, such that it destroys the
superconductivity state, we call it as vortices condensation. Otherwise, we
name it vortices dilution. In summary, in the dilution scenario, the system
stays as a perfect superconductor and in condensation scenario, free Maxwell
theory is recovery. The JT mechanism describes the dilution/condensation
process. The following partition function

$\displaystyle Z[j]=\sum_{K}\int{\cal D}A\int{\cal D}\theta\;e^{-\int
d^{4}x\left(\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{q^{2}m^{2}}{2}\left(A_{\mu}+\frac{1}{q}\partial_{\mu}\theta+\frac{2\pi}{q}K_{\mu}\right)^{2}\right)}e^{-ie\int
d^{4}x\;A_{\mu}j^{\mu}}$ (1)

describes the dilution/condensation process, where $K$ stands for the
vortices. Condensation can be described as the case where $K$ becomes a
continuous field. An integration over $K$ can be computed and the result is a
free Maxwell theory will be obtained. The partition function in Eq. (1) can be
derived from

$\displaystyle Z[j]=\sum_{K}\int{\cal D}A\int{\cal D}\theta\int{\cal
D}\eta\;e^{-\int
d^{4}x\left(\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-iq\left(A_{\mu}+\frac{1}{q}\partial_{\mu}\theta+\frac{2\pi}{q}K_{\mu}\right)\eta\right)}e^{-ie\int
d^{4}x\;A_{\mu}j_{\mu}}e^{-S_{\eta}}\,\,.$ (2)

From the simplest example, where $S_{\eta}=0$ in Eq. (2), the reason why $K$
represents a vortice is clear. In this case, $\eta$ is a Lagrange multiplier.
Thus the gauge field is restricted to flux filaments. The presence of
$S_{\eta}$ in Eq. (2) is connected to both the contact terms and the
topological dynamics of the system Braga:2016atp . The action for $\eta$ in
$S_{\eta}$, in Eq. (2), to obtain Eq. (1), is such that

$\displaystyle
S_{\eta}=\frac{1}{2m^{2}}\eta_{\mu}\eta^{\mu}+\frac{1}{2m^{2}m^{2}}\eta_{\mu}\Box\eta^{\mu}+\cdots\,\,,$
(3)

where $m$ is a massive parameter. We will show later that $m$ is linked with
the penetration length of the superconductor. In order to obtain Eq. (1), only
the first term needs to be considered. As it was pointed out in Braga:2016atp
, the expansion in Eq. (3) is considered because the goal was to describe the
low energy fluctuations of the field $\eta$. The action in Eq. (1) is the one
who describes a superconductor with penetration length $1/m$. Thus, as it was
stated before, $m$ is linked to the penetration length of the superconductor
sample.

##  III Low-energy effective Chern-Simons-Proca theory

From the previous section, it can be seen that if we consider the low energy
fluctuations of $\eta_{\mu}$, we are able to describe the electromagnetic
response in a superconductor. The goal of this section is to use the formalism
described before to show how an effective CSP theory can be obtained in the
low energy regime.

In Granado:2019bqk ; Braga:2016atp  the authors explored the derivative
expansion concerning Eq. (3). In Granado:2019bqk , it was shown that for the
case of high energy fluctuations of $\eta_{\mu}$, i.e., the action for
$\eta_{\mu}$ in Eq. (3) can be written as
$S_{\eta}=\eta_{\mu}\sum_{n}(\Box/m^{2})^{n}\eta^{\mu}$. In this paper, we
dealt with the lower energies fluctuations of $\eta_{\mu}$, but now we will
consider at least one derivative. This can be described in the following
manner

$\displaystyle S_{\eta}$ $\displaystyle=$
$\displaystyle\frac{1}{2m^{2}}\eta_{\mu}\eta^{\mu}+\frac{\xi}{2m^{2}}\eta_{\mu}(\epsilon^{\mu\nu\rho}\partial_{\rho})\eta_{\nu}+\cdots\,\,,$
(4)

where $\xi$ is a massive parameter. As it will be shown later, this parameter
is linked to the coupling constant for the CS term. From the dimensional
analysis, $\xi$ is proportional to $1/m$ and $\xi$ can be connected to a
superconductor penetration length. In fact, all the massive parameters in Eq.
(4) could be rewritten in terms of $\xi$. In comparison to Eq. (3), the new
term represents a CS dynamics for the $\eta$ field. This possibility was
considered before in Braga:2016atp  for the case of different $\eta$’s fields.
By integrating out $\eta_{\mu}$, as a result we have

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{3}x\Bigg{(}\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{q^{2}m^{2}}{(1-2{\xi}^{2}\partial^{2})}A^{\prime}_{\mu}A^{\prime\mu}+\frac{2qm^{2}\xi^{2}}{(1-2{\xi}^{2}\partial^{2})}A^{\prime}_{\mu}{\partial^{\nu}\partial^{\mu}}A^{\prime}_{\nu}$
(5) $\displaystyle+$
$\displaystyle\frac{qm^{2}}{(1-2{\xi}^{2}\partial^{2})}\xi\epsilon^{\mu\nu\rho}A^{\prime}_{\nu}\partial_{\rho}A^{\prime}_{\mu}\Bigg{)}\,\,.$

By means of the change of variable, explained in Appendix A, the action in Eq.
(5), using the substitution of ${\cal
H}\left({\Box}\right)^{-1/2}A^{\prime}_{\mu}\to\tilde{A}_{\mu}$ becomes

$\displaystyle S=\int d^{3}x\left(\frac{1}{4}F_{\mu\nu}{\cal
H}\left({\Box}\right)F^{\mu\nu}+\frac{2qm^{2}\xi^{2}}{2}A^{\prime}_{\mu}{\partial^{\nu}\partial^{\mu}}A^{\prime}_{\nu}+\theta\epsilon^{\mu\nu\rho}A^{\prime}_{\nu}\partial_{\rho}A^{\prime}_{\mu}+\frac{q^{2}m^{2}}{2}A^{\prime}_{\mu}A^{\prime\mu}\right)\,\,,$
(6)

where $\theta$ is defined by $\theta=qm^{2}\xi$. Thus, in the Lorentz gauge,
as a result we have a massive Maxwell-Podolsky-CS theory. The CS term here
resemble an axion term. A study involving this action will be addressed in a
future work. The goal in this letter is to investigate how the CSP theory can
emerge as an effective low energy theory. As it is known, as higher is the
order of the derivative, more relevant is the term in the ultraviolet regime
of the theory. Thus, in the low energy regime of the theory, the most relevant
terms are the ones who has one, or no derivatives. Therefore, in the Lorentz
gauge $\partial^{\mu}A_{\mu}=0$, from Eq. (6), we can see that the most
relevant terms in low energy limit are

$\displaystyle S_{eff}=\int
d^{3}x\left(\theta\epsilon^{\mu\nu\rho}A^{\prime}_{\nu}\partial_{\rho}A^{\prime}_{\mu}+\frac{q^{2}m^{2}}{2}A^{\prime}_{\mu}A^{\prime\mu}\right)\,\,,$
(7)

The action in Eq. (7) is precisely the one in Niemi:1994nw . In Niemi:1994nw ,
it was shown that Eq. (7) is precisely the topological CS quantum mechanical
model analyzed in djt .

##  IV Comments and conclusions

Through the last decades, the Abelian and non-Abelian versions of $D=3$ CS
theory have found several applications in several areas of physics. The gauge
invariant modification of the Abelian CS theory, discussed here, obtained by
adding a mass term to the gauge field was analyzed in Niemi:1994nw .

In this letter, we have used the JT approach to analyze a system whose low
energy degrees of freedom are described by the Abelian CSP theory. In our
approach, the CSP theory lives in the infrared regime of a massive Maxwell-
Podolsky-CS theory. In Granado:2019bqk , the authors use the JT mechanism to
obtain a massive Maxwell-Podolsky theory. In this paper it was pointed out
that, in the case of a massive Maxwell-Podolsky theory, the massive parameter
$m$ is still connected to the superconductor penetration length. Here we
obtain a massive Maxwell-Podolsky-CS theory. Due to the presence of the CS
term a careful analysis needs to be addressed and this will be presented in a
future work.

The procedure of introducing a massive Proca term brought together a new
dimensionfull parameter, which is an interesting perturbation of the CS action
in the infrared limit. As it was pointed in Niemi:1994nw , there could be
underlying consequences, for example, in the quantum Hall effect or high-
temperature superconductivity. As it was stated before, the JT mechanism was
used to describe regular superconductors and topological superconductors in
Braga:2016atp . As perspective of future research papers, we intend to
investigate the role of the CSP in high-temperature superconductivity
phenomena. This opens up a new road for the application the JT mechanism in
the superconductivity matters.

##  Appendix A Change of variables and Integration over $\eta$

In this Appendix we will review the change of variables introduced in
Granado:2019bqk . By means of a change of variables, the nonlocality included
in the potential term can be transferred to the kinetic term and vice-versa.
For instance, let us consider the following Lagrangian

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{m}\phi\,\text{exp}\,\Bigg{(}\frac{\Box}{2\mu^{2}}\Bigg{)}\partial^{m}\phi-V(\phi)\,\,.$
(8)

As it can be seen, the nonlocality is presented in the kinetic term. Let us do
the change of variables

$\displaystyle\text{exp}\Bigg{(}\frac{\Box}{2\mu^{2}}\Bigg{)}\,\phi\,\to\,\tilde{\phi}\,\,.$
(9)

It is clear that the Jacobian of this change of variable is
$e^{\frac{\Box}{2\mu^{2}}}\delta(x-y)|_{x=y}$, which is a constant whose
presence modifies trivially the generating functional. Besides, we assume that
this change of variables is invertible, i.e., it is performed only with the
fields defined in such a manner that a non-zero field, namely, a field which
is not equal identically to zero, is mapped to a non-zero field only. It
implies that if $\phi|_{x=y}=0$, i.e., $\phi$ vanishes at some surfaces only,
we have $e^{\frac{\Box}{2\mu^{2}}}\phi|_{x=y}=0$, and the same is true for
higher derivatives of $\phi$. In other words, if the field or some linear
combination of fields and its derivatives vanishes at some surface, its image
also will vanish on the same surface. Our Lagrangian takes the form

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{m}\tilde{\phi}\partial^{m}\tilde{\phi}-V\Big{(}e^{-\frac{\Box}{2\mu^{2}}}\,\tilde{\phi}\Big{)}.$
(10)

So, the potential becomes nonlocal instead of the kinetic term, just like in
certain cases when the kinetic term looks like $\phi\Box\widehat{T}\phi$, and
the potential term looks like $((\widehat{T})^{1/2}\phi)^{n}$, where
$\widehat{T}$ is an operator introducing the nonlocality.

Considering both terms in Eq. (4), we can solve the path integral for $\eta$
by using its equation of motion. The equation of motion for $\eta$ reads

$\displaystyle B_{\sigma\nu}\eta_{\nu}$ $\displaystyle=$ $\displaystyle-
iqm^{2}A^{\prime}_{\sigma}$ (11)

where

$B_{\sigma\nu}=\left(\delta_{\sigma\nu}+{\xi}\epsilon^{\sigma\nu\rho}\partial_{\rho}\right)$
(12)

In order to find the solution of Eq. (11), we have to compute the inverse
operator of Eq. (12). This operator is not invertible. In this way we have to
supplement Eq. (4) such that

$B_{\nu\sigma}=\left(\delta_{\nu\sigma}+{\xi}\epsilon^{\sigma\nu\rho}\partial_{\rho}+\left(1/\alpha-1\right)\partial_{\sigma}\partial_{\nu}\right)\,\,,$

and now $B_{\nu\sigma}$ is invertible. In order to show this we have that
$B_{\nu\sigma}B_{\sigma\delta}^{-1}=\delta_{\nu\delta}$. Let us construct an
ansatz

$B_{\sigma\delta}^{-1}=A\delta_{\sigma\delta}+B\partial_{\sigma}\partial_{\delta}+C\epsilon_{\sigma\delta\rho}\partial_{\rho}\,\,.$
(13)

From this we have the following set of equations

$\displaystyle A(1-2{\xi}^{2}\partial^{2})$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{{\xi}2C-A\left(1/\alpha-1\right)}{(1+\left(1/\alpha-1\right)\partial^{2})}$
$\displaystyle C$ $\displaystyle=$ $\displaystyle-A\xi\,\,.$

For $\alpha=1$, we have that Eq. (13) reads

$B_{\sigma\delta}^{-1}=\frac{1}{(1-{\xi}^{2}2\partial^{2})}\left(\delta_{\sigma\delta}-2\xi^{2}{\partial_{\sigma}\partial_{\delta}}-\xi\epsilon_{\sigma\delta\rho}\partial_{\rho}\right)\,\,.$
(14)

The solution of Eq. (11) reads

$\eta_{\nu}=\frac{-iqm^{2}}{(1-{\xi}^{2}2\partial^{2})}\left(\delta_{\sigma\delta}-2\xi^{2}{\partial_{\sigma}\partial_{\delta}}-\xi\epsilon_{\sigma\delta\rho}\partial_{\rho}\right)A^{\prime}_{\sigma}\,\,.$
(15)

## Acknowledgments

The authors thank CNPq (Conselho Nacional de Desenvolvimento Científico e
Tecnológico), Brazilian scientific support federal agency, for partial
financial support, grants numbers 301562/2019-9 (A.Yu.P.) and 406894/2018-3
(E.M.C.A.).

## References

  * (1) A. J. Niemi and V. V. Sreedhar, Phys. Lett. B 336, 381 (1994).
  * (2) G. V. Dunne, R. Jackiw and C. A. Trugenberger, Phys. Rev. D 41, 661 (1990).
  * (3) L. S. Grigorio, M. S. Guimaraes, R. Rougemont and C. Wotzasek, JHEP 1108, 118 (2011).
  * (4) J. R. Nascimento, A. Y. Petrov, C. Wotzasek and C. A. D. Zarro, Phys. Rev. D 89, 065030 (2014).
  * (5) P. R. Braga, D. R. Granado, M. S. Guimaraes and C. Wotzasek, Ann. Phys. 374, 1 (2016).
  * (6) C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, JHEP 1209, 091 (2012).
  * (7) D. R. Granado, A. J. G. Carvalho, A. Y. Petrov and P. J. Porfirio, EPL 129, 51001 (2020).
  * (8) G. Dunne, R. Jackiw and C. Trugenberger, Phys. Rev. D 41, 661 (1990).


# On Restricted Intersections and the Sunflower Problem

Jeremy Chizewer  Department of Combinatorics and Optimization, University of
Waterloo, Waterloo, ON, CA, N2L 3G1. Department of Computer Science, Princeton
University, Princeton, NJ 08544, and Research supported in part by the Fifty
Five Fund for Senior Thesis Research (Class 1955) Fund, Princeton University.
Email<EMAIL_ADDRESS>
###### Abstract

A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$
such that every element in $X$ is in no set, every set, or exactly one set.
Erdős and Rado [5] showed that a family of sets of size $n$ contains a
sunflower if there are more than $n!(r-1)^{n}$ sets in the family. Alweiss et
al. [1] and subsequently Rao [7] and Bell et al. [2] improved this bound to
$(O(r\log(n))^{n}$.

We study the case where the pairwise intersections of the set family are
restricted. In particular, we improve the best-known bound for set families
when the size of the pairwise intersections of any two sets is in a set $L$.
We also present a new bound for the special case when the set $L$ is the
nonnegative integers less than or equal to $d$ using the techniques of Alweiss
et al. [1].

##  1 Introduction

A _set family_ ${\mathcal{F}}$ over a finite set $X$ is a collection of
subsets of $X$. We say a set family is _$n$ -uniform_ if every set in the
family has size $n$.

######  Definition 1.1 (Sunflower).

An $r$-sunflower is a collection of sets $S_{1},\dots,S_{r}$ such that

$S_{i}\cap S_{j}=S_{1}\cap S_{2}\cap\cdots\cap S_{r}=K\text{ for all $i\neq
j$.}$

We call the set $K$ the _core_ and the sets $S_{i}\backslash K$ the _petals._

Erdős and Rado [5] originally referred to sunflowers as $\Delta$-systems and
proved that given an $n$-uniform set family $\mathcal{F}$ with
$|\mathcal{F}|=n!(r-1)^{n}$ there exists an $r$-sunflower contained in
$\mathcal{F}$. Sunflowers were renamed by Deza and Frankl in [4], and the term
sunflower is now more popular. The sunflower problem has been studied in
several papers including [5, 3, 4, 1, 7, 6].

Erdős and Rado conjectured that a stronger bound holds than the one they
proved in the initial paper.

######  Conjecture 1.2 (Erdős and Rado [5]).

Let ${\mathcal{F}}$ be an $n$-uniform set family. There exists some constant
$C=C(r)$ depending only on $r$ such that ${\mathcal{F}}$ contains an
$r$-sunflower whenever

$|{\mathcal{F}}|>C^{\,n}.$

Recently, Alweiss et al. [1], and subsequently Rao [7] and Bell et al. [2],
made progress toward this bound by showing there exists some constant $C$ such
that ${\mathcal{F}}$ contains an $r$-sunflower whenever
$|{\mathcal{F}}|>[Cr\log(n)]^{n}$. We study the Sunflower Problem with added
restrictions on the pairwise intersections.

######  Definition 1.3.

Let ${\mathcal{F}}$ be a set family. We call ${\mathcal{F}}$ an _$L$
-intersecting family_, if there exists a set $L\subset\mathbb{N}$ such that

$|F_{i}\cap F_{j}|\in L\textrm{ for every
}F_{i},F_{j}\in{\mathcal{F}},\,\textrm{with }i\neq j.$

The problem of sunflowers in $L$-intersecting set families was first studied
in [6]. They show that given an $L$-intersecting set family ${\mathcal{F}}$
with $|L|=s$, the family ${\mathcal{F}}$ contains a $3$-sunflower whenever
$|{\mathcal{F}}|>(n^{2}-n+1)8^{s-1}2^{(1+\sqrt{5}/5)n(s-1)}$. We improve their
bound, and extend the result to all $r\geq 3$ in the following theorem, which
is one of our main results.

######  Theorem 1.4.

Let ${\mathcal{F}}$ be an $L$-intersecting, $n$-uniform set family, for some
set $L\subset\mathbb{N}$ with ${|L|=s\geq 1}$. Let
$m=\max\\{r-1,n^{2}-n+1\\}$. Then ${\mathcal{F}}$ contains an $r$-sunflower
whenever

$|{\mathcal{F}}|>2^{n\log_{2}(s+1)+s\log_{2}(m)}.$

We also consider the special case where $L=\\{0,1,\dots,d\\}$ for some
$d\in\mathbb{N}$. In this case, we call the set family _$d$ -intersecting_.
Using the techniques of [1, 7] we achieve the following bound, our second main
result.

######  Theorem 1.5.

Let ${\mathcal{F}}$ be a $d$-intersecting, $n$-uniform set family. There
exists an absolute constant $C$ such that for every $r,n\geq 3$, the family
${\mathcal{F}}$ contains an $r$-sunflower whenever

$|{\mathcal{F}}|>(4r)^{n}[Cr\log(rd)]^{d}.$

######  Corollary 1.6.

For any $C>1$, there exists $c=c(r)>0$ depending only on $C$ and $r$ such that
if ${\mathcal{F}}$ is an $n$-uniform, $r$-sunflower free family with
$|{\mathcal{F}}|\geq(C4r)^{n}$ for $3\leq r\leq\log n$ then there exists
$F_{1},F_{2}\in{\mathcal{F}}$ such that

$|F_{1}\cap F_{2}|\geq cn/\log\log n.$

Corollary 1.6 follows immediately from Theorem 1.5 by setting $d=cn/\log\log
n$. The following question naturally arises.

######  Question 1.7.

Do there exist constants $C,c>0$ depending only on $r$ and possibly each other
such that if ${\mathcal{F}}$ is an $n$-uniform, $r$-sunflower free family of
size $|{\mathcal{F}}|\geq C^{\,n}$, then there exist
$F_{1},F_{2}\in{\mathcal{F}}$ such that $|F_{1}\cap F_{2}|\geq cn$?

Corollary 1.6 is of interest because the statement in Question 1.7 is
equivalent to Conjecture 1.2.

The remainder of the paper is organized as follows. Theorem 1.4 will be proved
in Section 2, and Theorem 1.5 in Section 3. We briefly discuss the results
Section 4.

##  2 Proof of Theorem 1.4

We proceed using similar ideas to the original proof of Erdős and Rado [5],
beginning with the following lemma.

######  Lemma 2.1 (Deza [3]).

Let ${\mathcal{F}}$ be an $L$-intersecting, $n$-uniform set family where
$L=\\{t\\}$ for some $0\leq t<n$. If ${\mathcal{F}}\geq n^{2}-n+2$ then
${\mathcal{F}}$ is a sunflower.

Lemma 2.1 provides the base case for our induction argument. We also state the
following definition, which will be useful in this section and the next one.

######  Definition 2.2.

Given a family ${\mathcal{F}}$ over $X$, and a set $T\subseteq X$ the “link”
of ${\mathcal{F}}$ at $T$, denoted ${\mathcal{F}}_{T}$, is defined as

${\mathcal{F}}_{T}=\\{F\,\backslash\,T:F\in{\mathcal{F}},T\subseteq F\\}.$

###### Proof of Theorem 1.4.

We prove using induction a slightly stronger statement than that of the
theorem.

######  Claim 2.3.

Let ${\mathcal{F}}$ be an $L$-intersecting, $n$-uniform set family for
$L=\\{\ell_{1},\dots,\ell_{s}\\}$ with
${0\leq\ell_{1}<\ell_{2}<\cdots<\ell_{s}<n}$ and $s\geq 1$. Let
$m=\max\\{r-1,n^{2}-n+1\\}$. Then ${\mathcal{F}}$ contains an $r$-sunflower
whenever

$\displaystyle|{\mathcal{F}}|>\frac{n!m^{s}}{(\ell_{1}+1)!(\ell_{2}-\ell_{1})!(\ell_{3}-\ell_{2})!\cdots(\ell_{s}-\ell_{s-1})!(n-\ell_{s}-1)!}.$
(1)

###### Proof of claim.

Fix $m$ as above. We proceed by induction on $s$. Indeed, for $s=1$, we apply
Lemma 2.1 to get the result immediately. Suppose the result holds for $0<j<s$,
and let ${\mathcal{F}}$ be an $n$-uniform, $L$-intersecting family (with $L$
as above) with $|L|=s\geq 2$. Suppose that ${\mathcal{F}}$ satisfies
Inequality (1). Let ${\mathcal{S}}\subseteq{\mathcal{F}}$ be a maximal subset
of ${\mathcal{F}}$ such that for every $S_{i},S_{j}\in{\mathcal{S}}$, if
$i\neq j$, then $|S_{i}\cap S_{j}|=\ell_{1}$. If $|{\mathcal{S}}|>m$ then, by
Lemma 2.1, ${\mathcal{S}}$ (and hence ${\mathcal{F}}$) contains an
$r$-sunflower. Thus, without loss of generality, we can assume that
$|{\mathcal{S}}|\leq m$. By maximality of ${\mathcal{S}}$, every set in
${\mathcal{F}}$ intersects at least one set of ${\mathcal{S}}$ in at least
$\ell_{1}+1$ elements. Let $S\in{\mathcal{S}}$ be the set which intersects the
most elements of ${\mathcal{F}}$ in at least $\ell_{1}+1$ elements, and let

${\mathcal{F}}^{\prime}=\\{F\in{\mathcal{F}}:|F\cap S|\geq\ell_{1}+1\\}.$

By the pigeonhole principle $|{\mathcal{F}}^{\prime}|\geq|{\mathcal{F}}|/m$.
There are ${n\choose\ell_{1}+1}$ subsets of $S$ of size $\ell_{1}+1$, and
every set in ${\mathcal{F}}^{\prime}$ contains at least one such subset, so
again by the pigeonhole principle there exists a set $S^{\prime}\subseteq S$
such that $|S^{\prime}|=\ell_{1}+1$ and the link at $S^{\prime}$ in
${\mathcal{F}}^{\prime}$ satisfies

$\displaystyle|{\mathcal{F}}^{\prime}_{S^{\prime}}|\geq\frac{|{\mathcal{F}}|}{m{n\choose\ell_{1}+1}}>\frac{(n-\ell_{1}-1)!m^{s-1}}{(\ell_{2}-\ell_{1})!(\ell_{3}-\ell_{2})!\cdots(\ell_{s}-\ell_{s-1})!(n-\ell_{s}-1)!}.$
(2)

Let $L^{\prime}=\\{\ell_{2}-\ell_{1}-1,\dots,\ell_{s}-\ell_{1}-1\\}$. We
observe that ${\mathcal{F}}^{\prime}_{S^{\prime}}$ is an
$(n-\ell_{1}-1)$-uniform, $L^{\prime}$-intersecting family, and
$|L^{\prime}|=s-1$. Thus, by Inequality (2) and induction
${\mathcal{F}}^{\prime}_{S^{\prime}}$ contains an $r$-sunflower (note that $m$
still satisfies the requirements of the induction hypothesis). Let
$F_{1},\dots,F_{r}\in{\mathcal{F}}^{\prime}_{S^{\prime}}$ be an $r$-sunflower,
then taking $F_{1}\cup S^{\prime},\dots,F_{r}\cup S^{\prime}\in{\mathcal{F}}$
gives an $r$-sunflower. ∎

Theorem 1.4 follows immediately from Claim 2.3 using the bound ${n\choose
m_{1},\dots,m_{k}}\leq k^{n}$ for multinomial coefficients. ∎

##  3 Proof of Theorem 1.5

We proceed using a similar argument to the main theorem of [1]. We start by
stating some definitions from [1], so that we may apply the results.

######  Definition 3.1.

We say that an $n$-uniform family ${\mathcal{F}}$ over $X$ is $\kappa$-spread
if $|{\mathcal{F}}|\geq\kappa^{n}$ and for all $T\subseteq X$ with $|T|\leq n$
we have $|{\mathcal{F}}_{T}|\leq\kappa^{-|T|}|{\mathcal{F}}|$

We introduce weight functions, so that we can deal with multiset families
(this is not strictly necessary, but it makes it easier to apply the results
of Alweiss et al. [1]).

######  Definition 3.2.

A function $\sigma:{\mathcal{F}}\to\mathbb{Q}$ is a weight function on a set
family ${\mathcal{F}}$ if it maps each set in ${\mathcal{F}}$ to a rational
weight, such that not all sets have weight zero. Moreover we define
$\sigma({\mathcal{S}})=\sum_{S\in{\mathcal{S}}}\sigma(S)$ for a set family
${\mathcal{S}}\subseteq{\mathcal{F}}$.

Our next definition generalizes the idea of $\kappa$-spread using weight
functions.

######  Definition 3.3.

We say that a set family ${\mathcal{F}}$ over $X$, and corresponding weight
function $({\mathcal{F}},\sigma)$, is $\mathbf{s}$-spread if
$\mathbf{s}=(s_{0};s_{1},\dots,s_{n})$ satisfies $s_{0}\geq
s_{1}\geq\cdots\geq s_{n}\geq 0$ with $\sigma({\mathcal{F}})\geq s_{0}$ and
for every set $T\subseteq X$ the subfamily
$\mathcal{T}=\\{F\in{\mathcal{F}}:T\subseteq F\\}$, satisfies
$\sigma(\mathcal{T})\leq s_{|T|}$.

Now we define $(\alpha,\beta)$-satisfying families, for the probabilistic
arguments that follow. We write $R\sim{\mathcal{U}}(X,\alpha)$ whenever
$R\subseteq X$ is generated by taking each element of $X$ uniformly and
independently at random with probability $0\leq\alpha\leq 1$.

######  Definition 3.4.

Let $0<\alpha,\beta<1$. A family ${\mathcal{F}}$ is
$(\alpha,\beta)$-satisfying if given $R\sim{\mathcal{U}}(X,\alpha)$,

${\mathbb{P}}_{R}(\exists S\in{\mathcal{F}},S\subseteq R)>1-\beta.$

Finally, we say that a family ${\mathcal{F}}$ is _$\mathbf{s}$ -spread_ if
there exists a weight function $\sigma$ such that $({\mathcal{F}},\sigma)$ is
$\mathbf{s}$-spread, and a weight profile $\mathbf{s}$ is _$(\alpha,\beta)$
-satisfying_ if any $\mathbf{s}$-spread family ${\mathcal{F}}$ is
$(\alpha,\beta)$-satisfying. Using these definitions, we can now state the
following lemmas.

######  Lemma 3.5 ([1, Lemma 1.6]).

If ${\mathcal{F}}$ is a $(1/r,1/r)$-satisfying family, and
$\emptyset\notin{\mathcal{F}}$, then ${\mathcal{F}}$ contains $r$ pairwise
disjoint sets.

######  Lemma 3.6 ([7, Lemma 4]).

Let $0<\alpha,\beta<1/2$. There exists a universal constant $C>1$ such that if
$\kappa=\kappa(n,\alpha,\beta)=C\log(n/\beta)/\alpha$ and a multiset family
${\mathcal{F}}$ over $X$ is a $\kappa$-spread, $n$-uniform family then
${\mathcal{F}}$ is $(\alpha,\beta)$-satisfying.

The next lemma, which is the main technical result of this section, will allow
us to use Lemma 3.6 on sets of size $d$ for a $d$-intersecting family.

######  Lemma 3.7.

Let ${\mathcal{F}}$ be a $d$-intersecting, $n$-uniform set family that is
$\mathbf{s}$-spread, such that
$\mathbf{s}:=(|{\mathcal{F}}|;s_{1},\dots,s_{d},1,\dots,1)$. Let $p,\delta>0$,
and suppose that
$\mathbf{s}^{\prime}=((1-\delta)|{\mathcal{F}}|;s_{1},\dots,s_{d})$ is
$(\alpha^{\prime},\beta^{\prime})$-satisfying. Then ${\mathcal{F}}$ is
$(\alpha,\beta)$-satisfying for

$\alpha=p+(1-p)\alpha^{\prime}\textrm{, and
\,}\beta=\beta^{\prime}+(2/p)^{n}/(\delta|{\mathcal{F}}|)$

Before proving Lemma 3.7, we define a notion of “good” and “bad” set pairs.
Bounding the number of bad pairs is the key idea in this proof.

######  Definition 3.8.

Let ${\mathcal{F}}$ be an $n$-uniform family over $X$ and let $W\subseteq X$.
Given $S\in{\mathcal{F}}$ and $w\in[n]$ we call the set pair $(W,S)_{w}$ good
if there exists a set $S^{\prime}\in{\mathcal{F}}$ (possibly with
$S^{\prime}=S$) such that

$S^{\prime}\,\backslash\,W\subseteq S\,\backslash\,W\text{, and
}|S^{\prime}\,\backslash\,W|\leq w$

We call $S^{\prime}$ a _witness_ to the goodness of $(W,S)_{w}$. We call a set
bad otherwise.

We use Definition 3.8 with $w=d$ for our purposes.

###### Proof of Lemma 3.7.

Let ${\mathcal{F}}$ be a $d$-intersecting, $n$-uniform set family over $X$,
with $|X|=x$. We begin by bounding the number of bad set pairs using an
encoding inspired by [1]. Suppose $(W,S)_{d}$ is a bad pair for $W\subseteq X$
where $|W|=px$ and $S\in{\mathcal{F}}$. First, we consider all possible sets
$W\cup S$. Since $|S|=n$ and $|W|=px$, we know that $px\leq|W\cup S|\leq
px+n$. Hence, there are

$\sum_{i=0}^{n}{x\choose
px+i}\leq\sum_{i=0}^{n}\left(\frac{1-p}{p}\right)^{i}{x\choose
px}\leq\left(\frac{1-p}{p}+1\right)^{n}{x\choose px}=p^{-n}{x\choose px}$

possible sets for $W\cup S$. Let $W\cup S$ be the first piece of information
in the encoding. There are $2^{n}$ possible values for $W\cap S$ since
$|S|=n$. Let $W\cap S$ be the second piece of information in the encoding. Now
we claim that given these two pieces of information and the additional
information that the corresponding set pair $(W,S)_{d}$ is bad, we can
reconstruct $(W,S)_{d}$. Indeed, if we knew $S$ in addition to this
information, we could clearly reconstruct $(W,S)_{d}$. Let
$S^{\prime}\in{\mathcal{F}}$ and suppose that $S^{\prime}\subseteq W\cup S$,
so that $S^{\prime}\,\backslash\,W\subseteq S\,\backslash\,W$. Since
$(W,S)_{d}$ is bad, it must be that $|S^{\prime}\,\backslash\,W|>d$, hence
$|S^{\prime}\cap S|>d$. Since ${\mathcal{F}}$ is $d$-intersecting,
$S^{\prime}=S$ and there is a unique set $S$ for a given $W\cup S$, which can
be computed by taking the unique set $S\subseteq W\cup S$. Since we also know
$W\cap S$, we can compute $W$. Therefore, there are at most
$(2/p)^{n}{x\choose px}$ bad set pairs. Since there are ${x\choose px}$
possible sets $W$, the expected number of bad set pairs for a given $W$ is
$(2/p)^{n}$. Let ${\mathcal{S}}(W)=\\{S\in{\mathcal{F}}:(W,S)_{d}\text{ is
bad}\\}$. By Markov’s inequality, the probability over $W$ drawn uniformly
from ${X\choose px}$, the set of subsets of $X$ with size $px$, satisfies

$\displaystyle{\mathbb{P}}_{W}(|{\mathcal{S}}(W)|\geq\delta|{\mathcal{F}}|)\leq\frac{(2/p)^{n}}{\delta|{\mathcal{F}}|}$
(3)

When $|{\mathcal{S}}(W)|\leq\delta|{\mathcal{F}}|$ we define a new $d$-uniform
multiset family ${\mathcal{F}}^{\prime}$ over $X\,\backslash\,W$ which is
$\mathbf{s}^{\prime}=((1-\delta)|{\mathcal{F}}|;s_{1},\dots,s_{d})$-spread.
The rest of the proof follows immediately from the arguments in Section 2.1 of
[1]. ∎

###### Proof of Theorem 1.5.

We roughly follow the argument used in [1, 7]. Let ${\mathcal{F}}$ be a
$d$-intersecting, $n$-uniform family over $X$ of size
$|{\mathcal{F}}|>(4r)^{n}[Cr\log(rd)]^{d}$, for $C$ to be chosen later. Let
$T\subseteq X$ be the largest set with $|T|\leq d$ (possibly $|T|=0$) such
that

$|{\mathcal{F}}_{T}|\geq[Cr\log(rd)]^{-|T|}|{\mathcal{F}}|.$

We claim that ${\mathcal{F}}_{T}$ is $\kappa=Cr\log(rd)$-spread. Indeed, if
$|T|=d$ then ${\mathcal{F}}_{T}$ is a family of pairwise disjoint sets, and
otherwise we can find a link at $T^{\prime}\subseteq X\,\backslash\,T$ such
that $|T^{\prime}|>0$ and

$({\mathcal{F}}_{T})_{T^{\prime}}\geq[Cr\log(rd)]^{-|T^{\prime}|}|{\mathcal{F}}_{T}|.$

But then, taking $T^{\prime}\cup T$ gives a larger set with

$|{\mathcal{F}}_{T\cup T^{\prime}}|\geq[Cr\log(rd)]^{-|T\cup
T^{\prime}|}|{\mathcal{F}}|,$

a contradiction. Hence, ${\mathcal{F}}_{T}$ is $\mathbf{s}$-spread and
$(d-|T|)$-intersecting for weight profile

${\mathbf{s}}=(|{\mathcal{F}}_{T}|;|{\mathcal{F}}_{T}|/\kappa,\dots,|{\mathcal{F}}_{T}|/\kappa^{d-|T|},1,\dots,1),$

taking $\sigma(F)=1$ for all $F\in{\mathcal{F}}_{T}$. Let
$\mathbf{s}^{\prime}=(|{\mathcal{F}}_{T}|/2;|{\mathcal{F}}_{T}|/\kappa,\dots,|{\mathcal{F}}_{T}|/\kappa^{d-|T|})$.
As in [1], we observe that if a family is
$(|{\mathcal{F}}_{T}|/2;|{\mathcal{F}}_{T}|/\kappa,\dots,|{\mathcal{F}}_{T}|/\kappa^{d-|T|})$-spread,
then it is also
$\mathbf{s}^{\prime\prime}=(|{\mathcal{F}}_{T}|;|{\mathcal{F}}_{T}|/\kappa^{\prime},\dots,|{\mathcal{F}}_{T}|/\kappa^{\prime(d-|T|)})$-spread
for $\kappa^{\prime}=\kappa/2$. By Lemma 3.6, $\mathbf{s}^{\prime\prime}$ (and
hence also $\mathbf{s}^{\prime}$) is $(\frac{1}{2r},\frac{1}{2r})$-satisfying
for $C$ chosen sufficiently large. Hence, by Lemma 3.7 with $\delta=1/2$ and
$p=\frac{1}{2r}$, we know that $\mathbf{s}$ is $(1/r,1/r)$-satisfying choosing
$C$ sufficiently large according to the result of the lemma. Therefore, by
Lemma 3.5, this implies ${\mathcal{F}}_{T}$ contains $r$ pairwise disjoint
sets. Let $F_{1},\dots,F_{r}\in{\mathcal{F}}_{T}$ be pairwise disjoint. Then
$F_{1}\cup T,\dots,F_{r}\cup T\in{\mathcal{F}}$ is an $r$-sunflower. ∎

##  4 Discussion

It is easy to see that Theorem 1.5 implies a stronger statement than the best
known bound whenever $d=o(n)$. The original bound of Erdős and Rado can be
directly applied to $d$-intersecting sets families to achieve a bound of
$(r-1)^{d+1}n!/(n-d)!$. A natural question is when Theorem 1.5 improves this
trivial bound? There is some constant $c>0$ such that

$\displaystyle\frac{(r-1)^{d+1}n!}{(n-d)!}\geq\left[\frac{(r-1)n}{2}\right]^{cn/\log\log
n}\geq(4r)^{n}[Cr\log(rcn/\log\log n)]^{cn/\log\log n}$

for $d\geq cn/\log\log n$ and $n$ sufficiently large. Hence, in this regime,
we improve the best known bound. In particular, this motivates Corollary 1.6
and Question 1.7.

Acknowledgement: We thank Ryan Alweiss, Noga Alon, Robert Sedgewick and
Stephen Melczer for helpful suggestions, and to Lutz Warnke for telling us
about [2]. Thank you to The Fifty Five Fund for Senior Thesis Research (Class
1955) Fund, Princeton University for partially supporting this research.

## References

  * [1] R. Alweiss, S. Lovett, K. Wu, J. Zhang, Improved bounds for the sunflower lemma, Ann. of Math. (2) 194 (2021), no. 3, 795–815
  * [2] T. Bell, S. Chueluecha and L. Warnke, Note on sunflowers, Discrete Mathematics, 344 (2021), no. 7, 112367
  * [3] M. Deza, Solution d’un problème de Erdős-Lovász, J. Comb. Theory, Ser. B 16 (2), 1974, 166–167.
  * [4] M. Deza and P. Frankl. Every large set of equidistant $(0,+1,-1)$-vectors forms a sunflower. Combinatorica, 1(3):225–231, 1981.
  * [5] P. Erdős and R. Rado. Intersection theorems for systems of sets. Journal of the London Mathematical Society, 35(1):85-90, 1960.
  * [6] G. Hegedűs, Sunflowers and $L$-intersecting families, AKCE International Journal of Graphs and Combinatorics, 17:1, 402-406, 2019.
  * [7] A. Rao, Coding for Sunflowers, Discrete Analysis, 2020:2.


# Brownian oscillator with time-dependent strength: A delta function protocol

Alex V. Plyukhin<EMAIL_ADDRESS>Department of Mathematics, Saint Anselm
College, Manchester, New Hampshire 03102, USA

###### Abstract

We consider a classical Brownian oscillator of mass $m$ driven from an
arbitrary initial state by varying the stiffness $k(t)$ of the harmonic
potential according to the protocol $k(t)=k_{0}+a\,\delta(t)$, involving the
Dirac delta function. The microscopic work performed on the oscillator is
shown to be $W=(a^{2}/2m)\,q^{2}-aqv$, where $q$ and $v$ are the coordinate
and velocity in the initial state. If the initial distribution of $q$ and $v$
is the equilibrium one with temperature $T$, the average work is $\langle
W\rangle=a^{2}T/(2m\,k_{0})$ and the distribution $f(W)$ has the form of the
product of exponential and modified Bessel functions. The distribution is
asymmetric and diverges as $W\to 0$. The system’s response for $t>0$ is
evaluated for specific models.

##  I Introduction

Brownian motion in a time dependent harmonic potential is a stochastic process
important from both theoretical and experimental points of view. It provides
an insightful example and testing ground for subtle theoretical results, like
fluctuation theorems PP . It is also directly relevant to important
experimental techniques to study mesoscopic systems using optical traps. In
such experiments, the system is driven from thermal equilibrium by varying the
strength of the trap, i.e. the stiffness of the harmonic potential $k(t)$,
according to a certain protocol, and the work $W$ on the system, as well as
the system’s response, are recorded traps1 ; traps2 .

The optical trap setup has been used also to design Brownian engines with a
single optically trapped Brownian particle as a working substance and with the
trap stiffness $k(t)$ as a control parameter Blickle ; Martinez ; SM ; HR ;
Ryabov_review . Such machines often operate out of equilibrium using driving
protocols of finite duration and thus belong to the realm of finite-time
thermodynamics FT ; Watanabe .

The steps of isothermal compression (expansion) of the macroscopic Carnot
cycle correspond in Brownian engines to the increase (decrease) of the
stiffness $k(t)$ at fixed temperature during a finite switching time interval
$(0,t_{s})$. The corresponding work $W$ strongly fluctuates and is governed by
equations of stochastic dynamics of Langevin or Fokker-Planck types with the
time-dependent potential $V(q,t)=k(t)q^{2}/2$. The microscopic work can be
written as Jar

$\displaystyle
W=\int_{0}^{t_{s}}\frac{dV(q,t)}{dk}\,\dot{k}(t)\,dt=\frac{1}{2}\,\int_{0}^{t_{s}}q^{2}(t)\,\dot{k}(t)\,dt,$
(1)

and the average work is a functional of the mean-square displacement $\langle
q^{2}(t)\rangle$,

$\displaystyle\langle W\rangle=\frac{1}{2}\,\int_{0}^{t_{s}}\langle
q^{2}(t)\rangle\,\dot{k}(t)\,dt.$ (2)

Since the microscopic work is a nonlinear functional of $q$, the work
distribution is not Gaussian in general. For overdamped Langevin dynamics with
white Gaussian noise the asymptotic distribution for large absolute values of
work was found to have the form

$\displaystyle f(W)=c_{1}\frac{e^{-c_{2}|W|}}{\sqrt{|W|}},$ (3)

where $c_{1},c_{2}$ depend on the specific form of the protocol $k(t)$ Engel ;
NE . This form, with an exponential tail and a power-law prefactor, was found
for $f(W)$ also for Langevin dynamics beyond the overdamped limit Kwon . On
the other hand, the central part of the work distribution was predicted to
have an approximately Gaussian form in the limit of a slow driving Seifert ;
Speck .

The evaluation of the work distribution in closed form for an arbitrary
protocol and for the whole range of $W$ appears to be a rather formidable
task. However, for a special protocol, often referred to as the instantaneous
quench PP , the calculations are very simple. In that protocol at $t=0$ the
stiffness is instantaneously switched from $k_{0}$ to a new value $k_{1}$,
i.e.

$\displaystyle k(t)=k_{0}+(k_{1}-k_{0})\,\theta(t),$ (4)

where $\theta(t)$ is the step function. The corresponding microscopic work on
the system is equal to the difference of the system’s potential energy

$\displaystyle W=\frac{k_{1}-k_{0}}{2}\,q^{2},$ (5)

where $q=q(0)$. Because the protocol’s duration is effectively zero,
statistics of $W$ does not depend on details of stochastic dynamics (such as
statistics of the noise) and is completely determined by the distribution of
the oscillator’s initial coordinate $q$. Suppose at $t<0$ the oscillator with
the stiffness $k=k_{0}$ is in thermal equilibrium and the coordinate and
velocity are distributed according to the canonical distribution

$\displaystyle\rho_{0}(q,v)=\frac{1}{Z}\,\exp\left(-\frac{m\,v^{2}}{2T}-\frac{k_{0}\,q^{2}}{2T}\right).$
(6)

Here and below temperature is in energy units. Taking the average of Eq. (5)
with distribution (6) gives the mean work

$\displaystyle\langle W\rangle=\frac{k_{1}-k_{0}}{k_{0}}\,\frac{T}{2}.$ (7)

This result also follows from Eqs. (2) and (4). According to Eqs. (5)-(7), the
normalized work $x=W/\langle W\rangle$ is a square of the standard normal
random variable $z=\sqrt{k_{0}/T}q$, $x=z^{2}$. Therefore $x$ is distributed
with the chi-squared $\chi_{1}^{2}$ distribution Simon

$\displaystyle
f(x)=\frac{1}{\sqrt{2\pi}}\frac{e^{-x/2}}{\sqrt{x}}\,\theta(x),$ (8)

and the distribution for $W=x\,\langle W\rangle$ has the form

$\displaystyle f(W)=\frac{1}{\sqrt{2\pi\,\langle
W\rangle\,W}}\,\exp\left(-\frac{W}{2\langle
W\rangle}\right)\,\theta\left(\frac{W}{\langle W\rangle}\right).$ (9)

This result corroborates Eq. (3), but in contrast to the latter it holds for
any $W$. Thus for the instantaneous quench the work distribution is non-
Gaussian for any values of $W$. The distribution is exactly zero when $W$ has
a sign opposite to that of $\langle W\rangle$.

For the instantaneous quench the validity of the Jarzynski equality Jar

$\displaystyle\langle e^{-W/T}\rangle=e^{-\Delta F/T}$ (10)

can be directly verified. For the oscillator with stiffness $k$ in thermal
equilibrium the free energy is $F(k)=-T\,\ln(2\pi T\sqrt{m/k})$. Then $\Delta
F=F(k_{1})-F(k_{0})=-T\,\ln(\sqrt{k_{0}/k_{1}})$, and the right-hand side of
Eq. (10) equals $\sqrt{k_{0}/k_{1}}$. The average in the left-hand side can be
taken either over $q,v$ with distribution (6), or over $W$ with distribution
(9):

$\displaystyle\langle e^{-W/T}\rangle=\iint
exp\left(-\frac{k_{1}-k_{0}}{2T}\,q^{2}\right)\,\rho_{0}(q,v)\,\,dq\,dv=\int\exp\left(-\frac{W}{T}\right)\,f(W)\,dW=\sqrt{\frac{k_{0}}{k_{1}}}.$
(11)

The purpose of this paper is to consider another limiting and idealized case
when the stiffness is perturbed according to the protocol

$\displaystyle k(t)=k_{0}+a\,\delta(t),$ (12)

where $\delta(t)$ is the Dirac delta function and the intensity of the
perturbation $a$, which may be of any sign, has units of mass/time. We shall
refer to Eq. (12) as the delta protocol. Of course, the delta protocol cannot
be practically realized per se and should be viewed as an asymptotic form of
spike-like perturbations with a high amplitude and a short duration. We shall
see that the properties of the the delta protocol in many respects differ from
that for the instantaneous quench.

##  II Work

In the sections to follow we show that for the delta protocol (12) the
microscopic work on the system is

$\displaystyle W(q,v)=\frac{a^{2}}{2m}\,q^{2}-a\,q\,v,$ (13)

where $q=q(0^{-})$ and $v=v(0^{-})$ are initial coordinate and velocity just
before the perturbation is applied. Similar to the instantaneous quench, see
Eq. (5), the statistics of $W$ is completely determined by that of the initial
state, but for the delta protocol it depends on both initial coordinate and
velocity. The average work is governed by the mean square displacement
$\langle q^{2}\rangle$ and the position-velocity correlation $\langle
q\,v\rangle$ in the initial state.

Let us assume that for $t<0$ the system is in thermal equilibrium with
distribution (6). In that case the initial correlation $\langle qv\rangle$ is
zero, and the average work is determined by the initial mean square
displacement $\langle q^{2}\rangle$ only,

$\displaystyle\langle W\rangle=\iint
W(q,v)\,\rho_{0}(q,v)\,dq\,dv=\frac{a^{2}}{2m}\langle
q^{2}\rangle=\frac{a^{2}T}{2mk_{0}}=\frac{\alpha^{2}}{2}\,T,\qquad\alpha=\frac{a}{m\,\omega_{0}}.$
(14)

Here $\omega_{0}=\sqrt{k_{0}/m}$ is the oscillator’s frequency, and $\alpha$
is the characteristic dimensionless parameter of the protocol. One observes
that $W$ and $\langle W\rangle$ may be, in contrast to the instantaneous
quench, of different signs. Using Eq. (13), one can readily evaluate the
moment-generating function for the dimentionless work $w=W/T$,

$\displaystyle M_{w}(t)=\langle e^{tw}\rangle=\iint
e^{t\,W(q,v)/T}\,\rho_{0}(q,v)\,dq\,dv=\frac{1}{\sqrt{1-\alpha^{2}(t+t^{2})}}.$
(15)

Then the moments of $W$ can be evaluated as

$\displaystyle\mu_{n}=\langle
W^{n}\rangle=T^{n}\,\frac{d^{n}}{dt^{n}}\,M_{w}(t)\bigg{\rvert}_{t=0}.$ (16)

For the first few moments one obtains

$\displaystyle\mu_{1}$ $\displaystyle=$
$\displaystyle\mu=\frac{1}{2}\,\alpha^{2}\,T,$ $\displaystyle\mu_{2}$
$\displaystyle=$
$\displaystyle\left(\alpha^{2}+\frac{3}{4}\,\alpha^{4}\right)\,T^{2}=2\,T\,\mu+3\,\mu^{2},$
$\displaystyle\mu_{3}$ $\displaystyle=$
$\displaystyle\left(\frac{9}{2}\,\alpha^{4}+\frac{15}{8}\,\alpha^{6}\right)\,T^{3}=18\,T\,\mu^{2}+15\,\mu^{3},$
$\displaystyle\mu_{4}$ $\displaystyle=$
$\displaystyle\left(9\,\alpha^{4}+\frac{45}{2}\,\alpha^{6}+\frac{105}{16}\,\alpha^{8}\right)T^{4}=36\,T^{2}\,\mu^{2}+180\,T\,\mu^{3}+105\,\mu^{4},$
(17)

and the variance $\sigma^{2}=\mu_{2}-\mu^{2}$ equals

$\displaystyle\sigma^{2}=\left(\alpha^{2}+\frac{1}{2}\,\alpha^{4}\right)\,T^{2}=2T\mu+2\mu^{2}.$
(18)

From the thermodynamics point of view, the delta protocol is a cycle process,
for which the difference of the free energy is zero, $\Delta F=0$, and the
Jarzynski equality (10) takes the form $\langle e^{-W/T}\rangle=1$. The
validity of this result for the delta protocol is immediately obvious from Eq.
(15),

$\displaystyle\langle e^{-W/T}\rangle=M_{w}(-1)=1.$ (19)

For the dimensionless work $w=W/T$, the moment-generating function taken at
$-t$ is a bilateral (two-sided) Laplace transform $\mathcal{B}$ of the
distribution function (probability density function) $f(w)$,

$\displaystyle
M_{w}(-t)=\int_{-\infty}^{\infty}e^{-tw}f(w)\,dw=\mathcal{B}\\{f(w)\\}.$ (20)

Then from Eqs. (15) and (20), one finds

$\displaystyle\mathcal{B}\\{f(w)\\}=\frac{1}{\sqrt{1+\alpha^{2}(t-t^{2})}}.$
(21)

The inversion of this relation yields (see the Appendix)

$\displaystyle
f(w)=\frac{1}{\pi\,|\alpha|}\,\exp\left(\frac{w}{2}\right)\,K_{0}\left(\frac{\Delta}{2}\,|w|\right),$
(22)

where $K_{0}(x)$ is the modified Bessel function, the parameter $\Delta$ is

$\displaystyle\Delta=\sqrt{\frac{4}{\alpha^{2}}+1},$ (23)

and $\alpha$ is defined by Eq. (14). The distribution function for the work
$W=T\,w$ takes the form

$\displaystyle
f(W)=\frac{1}{\pi\,|\alpha|\,T}\,\exp\left(\frac{W}{2T}\right)\,K_{0}\left(\frac{\Delta}{2T}\,|W|\right),$
(24)

which is shown in Fig. 1. As for the instantaneous quench, the work
distribution diverges as $W\to 0$. This divergence does not lead to any
singularities of measurable quantities. In particular, using $f(W)$ to
evaluate the moments $\mu_{n}=\int f(W)W^{n}dW$, one recovers Eqs. (17). Also,
the validity of the Jarzynski equality, evaluated with $f(W)$, can be directly
verified:

$\displaystyle\langle e^{-W/T}\rangle=\int e^{-W/T}\,f(W)\,dW=\int
e^{-w}\,f(w)\,dw=1.$ (25)

Figure 1: The work distribution function (24) for several values of
$|\alpha|=|a|/(m\,\omega_{0})$.

Another remarkable, though quite expected, in view of Eq. (13), feature of the
distribution $f(W)$ is that it is asymmetric. Taking into account the
asymptotic form of the modified Bessel function
$K_{0}(x)\approx\sqrt{\pi/(2x)}\,e^{-x}$ for large $x$, one finds that the
asymptotic tails of the distribution for $|W|>2T/\Delta$ have the forms

$\displaystyle
f(W)\approx\frac{1}{\alpha\sqrt{\Delta\,\pi\,T\,W}}\exp\left(-\frac{\Delta-1}{2T}\,W\right),\quad
W>0$ (26)

and

$\displaystyle
f(W)\approx\frac{1}{\alpha\sqrt{\Delta\,\pi\,T\,|W|}}\exp\left(\frac{\Delta+1}{2T}\,W\right),\quad
W<0.$ (27)

The distribution decays faster for negative values of $W$. If in addition
$|\alpha|\gg 1$, then $\Delta\to 1$, $\Delta-1\to 2/\alpha^{2}$, and the above
expressions are further simplified:

$\displaystyle f(W)\approx\frac{1}{\sqrt{2\,\pi\,\langle
W\rangle\,W}}\exp\left(-\frac{W}{2\langle W\rangle}\right),\quad W>0,$ (28)

and

$\displaystyle f(W)\approx\frac{1}{\sqrt{2\,\pi\,\langle
W\rangle\,|W|}}\exp\left(\frac{W}{T}\right),\quad W<0.$ (29)

Note that the asymptotic expression (28) coincides with the exact distribution
for the instantaneous quench, Eq. (9).

Whereas the derivation of the above results is quite straightforward, there is
a subtlety related to the fact that for the delta protocol the microscopic
velocity is discontinuous at $t=0$, see Eq. (51) below. The next section
elaborates on that.

##  III Preliminaries

For the delta protocol (12) Eq. (2) for the average work takes the form

$\displaystyle\langle W\rangle=\frac{a}{2}\,\int_{-\epsilon}^{\epsilon}\langle
q^{2}(t)\rangle\,\dot{\delta}(t)\,dt,$ (30)

with $\epsilon>0$. Integrating by parts yields

$\displaystyle\langle
W\rangle=-\frac{a}{2}\,\int_{-\epsilon}^{\epsilon}\\!\delta(t)\,\frac{d}{dt}\,\langle
q^{2}(t)\rangle\,dt.$ (31)

As a next step, it might be tempting to apply the defining property of the
delta function

$\displaystyle\int_{-\epsilon}^{\epsilon}\delta(t)f(t)dt=f(0)$ (32)

to obtain

$\displaystyle\langle W\rangle=-\frac{a}{2}\,\,\frac{d}{dt}\,\langle
q^{2}(t)\rangle\bigg{\rvert}_{t=0}=-a\,\langle q(0)\,v(0)\rangle.$ (33)

Suppose the system is in thermal equilibrium at $t<0$, so that the position-
velocity correlation $\langle q(t)\,v(t)\rangle$ is zero at $t=0^{-}$.
Assuming that the correlation is continuous at $t=0$, one would find from Eq.
(33) that the mean work is zero, $\langle W\rangle=0$.

That conclusion is incorrect because, as we shall see below, the function

$\displaystyle f(t)=\frac{d}{dt}\,\langle q^{2}(t)\rangle=2\,\langle
q(t)\,v(t)\rangle$ (34)

is discontinuous at $t=0$. Since $f(0)$ is not well-defined, the property (32)
is ambiguous and the transition from Eq. (31) to Eq. (33) is in general
erroneous. The standard way to extend the property (32) for the case of
discontinuous $f(t)$ is to replace $f(0)$ by the average of $f(t)$ at the
discontinuity

$\displaystyle\int_{-\epsilon}^{\epsilon}\delta(t)f(t)dt=\frac{1}{2}\left[f(0^{-})+f(0^{+})\right].$
(35)

In our case $f(0^{-})=0$ (if the system is in equilibrium at $t<0$) and
$f(0^{+})$ will be shown below, see Eq. (60), to be

$\displaystyle f(0^{+})=-\frac{2aT}{mk_{0}}.$ (36)

Then from Eqs. (31) and (35) one gets

$\displaystyle\langle
W\rangle=-\frac{a}{4}\,f(0^{+})=\frac{a^{2}}{m\,k_{0}}\,\frac{T}{2},$ (37)

which is the result (14).

The ansatz (35) is known to be inconsistent in certain cases and ought to be
used with care GW . Below we consider the problem with two methods. The first
method is based on using ansatz (35). The second method does not rely on
ansatz (35), but involves a specific representation of $\delta(t)$ as a limit
of a rectangular impulse. We find that the two methods lead to the same
results.

##  IV Method I

According to Eq. (1), for the delta protocol the microscopic work is

$\displaystyle
W=\frac{a}{2}\,\int_{-\epsilon}^{\epsilon}\\!q^{2}(t)\,\dot{\delta}(t)\,dt=-a\int_{-\epsilon}^{\epsilon}\\!q(t)\,v(t)\,\delta(t)\,dt.$
(38)

Anticipating $q(t)$ and $v(t)$ to be, respectively, continuous and
discontinuous at $t=0$, and applying ansatz (35) one gets

$\displaystyle W=-\frac{a}{2}\,q(0)\,[v(0^{+})+v(0^{-})].$ (39)

Here $q=q(0)$ and $v=v(0^{-})$ are drawn from the initial distribution, which
may be, but not necessarily, the equilibrium distribution (6). In order to
find $v(0^{+})$ we need to solve equations of stochastic dynamics, find $v(t)$
for $t>0$, and take the limit $t\to 0^{+}$.

To this end, we assume that for $t>0$ the system is described by the
generalized Langevin equation Zwanzig ; Weiss

$\displaystyle\\!\\!m\,\ddot{q}(t)\\!=\\!-k(t)\,q(t)\\!-\\!m\int_{0}^{t}\\!K(t-\tau)\,\dot{q}(\tau)\,d\tau+\xi(t).$
(40)

Here $\xi(t)$ is the stationary fluctuating force (“noise”) which is zero-
centered, $\langle\xi(t)\rangle=0$, and related to the dissipation kernel
$K(t)$ via the fluctuation-dissipation relation

$\displaystyle\langle\xi(t)\,\xi(t^{\prime})\rangle=m\,T\,K(|t-t^{\prime}|).$
(41)

We do not assume that $\xi(t)$ is Gaussian. The anticipated results (13) and
(14) do not depend on the form of the dissipation kernel $K(t)$. Therefore,
instead of the generalized Langevin equation one can use its more simple and
familiar Markovian form with $K(t)=\gamma\,\delta(t)$. Yet we prefer to work
with the Langevin equation of the form (40) which is more general and, in
contrast to the Markovian counterpart, does not imply coarse-graining of time.
We shall show in Conclusion that the overdamped approximation (when the
acceleration term is put to zero) is inadequate to describe the delta
protocol.

The Langevin equation (40) for protocol (12) takes the form

$\displaystyle\ddot{q}(t)=-\left[\omega_{0}^{2}+\frac{a}{m}\,\delta(t)\right]\,q(t)-\int_{0}^{t}K(t-\tau)\,\dot{q}(\tau)\,d\tau+\frac{1}{m}\xi(t),$
(42)

where $\omega_{0}^{2}=k_{0}/m$ is the oscillator’s natural frequency. The
equation is linear and can be readily solved using Laplace transforms

$\displaystyle\mathcal{L}\\{f(t)]=\tilde{f}(s)=\int_{0^{-}}^{\infty}e^{-st}f(t)\,dt.$
(43)

Assuming that $q(t)$ (but not $v(t)=\dot{q}(t)$) is continuous at $t=0$, one
finds

$\displaystyle
q(t)=S_{0}(t)\,q(0)+G_{0}(t)\left[v(0^{-})\\!-\\!\frac{a}{m}q(0)\right]+\frac{1}{m}\\{G_{0}*\xi\\}.$
(44)

Here the asterisk in the last term stands for the convolution
$\\{G_{0}*\xi\\}=\int_{0}^{t}G_{0}(t-\tau)\,\xi(\tau)\,d\tau$, and the
relaxation functions $G_{0}(t)$ and $S_{0}(t)$ are defined in the Laplace
domain as follows,

$\displaystyle\tilde{G}_{0}(s)=\frac{1}{s^{2}+s\tilde{K}+\omega_{0}^{2}},\quad\tilde{S}_{0}(s)=\frac{s+\tilde{K}}{s^{2}+s\tilde{K}+\omega_{0}^{2}}=\frac{1}{s}\,\left[1-\omega_{0}^{2}\,\tilde{G}_{0}(s)\right].$
(45)

It is also convenient to use another relaxation function $R_{0}(t)$, which is
connected to $G_{0}(t)$ in the Laplace domain as

$\displaystyle\tilde{R}_{0}(s)=s\,\tilde{G}_{0}(s).$ (46)

One can show that the relaxation functions coincide with the normalized
correlations for the oscillator with $k=k_{0}$ in thermal equilibrium Wang ;
DV ; Goychuk ; Plyukhin

$\displaystyle\langle
q(t)q(t^{\prime})\rangle_{eq}=\frac{T}{k_{0}}\,S_{0}(t-t^{\prime}),\quad\langle
v(t)v(t^{\prime})\rangle_{eq}=\frac{T}{m}\,R_{0}(t-t^{\prime}),\quad\langle
q(t)v(t^{\prime})\rangle_{eq}=\frac{T}{m}\,G_{0}(t-t^{\prime}).$ (47)

From these relations it is clear that the relaxation functions are continuous
at $t=0$, and their initial values are

$\displaystyle S_{0}(0)=R_{0}(0)=1,\quad
G_{0}(0)=0,\quad\dot{S}_{0}(0)=\dot{R}_{0}(0)=0,\quad\dot{G}_{0}(0)=1.$ (48)

These initial values can also be obtained directly using the definition of the
relaxation functions in the Laplace domain, Eqs. (45) and (46).

Using the above relations, one finds that in the time domain the relaxation
functions are connected as follows

$\displaystyle\dot{G}_{0}(t)=R_{0}(t),\quad\dot{S}_{0}(t)=-\omega_{0}^{2}G_{0}(t).$
(49)

Then differentiating Eq. (44) yields

$\displaystyle
v(t)=-q(0)\,\left[\omega_{0}^{2}G_{0}(t)+\frac{a}{m}R_{0}(t)\right]+v(0^{-})\,R_{0}(t)+\frac{1}{m}\,\\{R_{0}*\xi\\}.$
(50)

Taking in Eqs. (44) and (50) the limit $t\to 0^{+}$ and taking into account
Eq. (48) one finds that at $t=0$ the coordinate $q(t)$ is continuous (in
accordance with our initial assumption), while the velocity $v(t)$ has a jump
discontinuity (unless $q(0)=0$):

$\displaystyle v(0^{+})=v(0^{-})-q(0)\,\frac{a}{m}.$ (51)

This relation can also be obtained directly by integrating the Langevin
equation (42) over $t$ from $0^{-}$ to $0^{+}$. Substituting Eq. (51) into Eq.
(39) yields for the microscopic work the result (13),

$\displaystyle W=\frac{a^{2}}{2m}\,q^{2}-a\,q\,v,$ (52)

where $q=q(0)$ and velocity $v=v(0^{-})$ are the coordinate and velocity
immediately before the perturbation.

##  V Energy

Suppose the system at $t<0$ is in thermal equilibrium with the distribution
$\rho_{0}$, Eq. (6). For the instantaneous quench $k_{0}\to k_{1}$ at $t=0$
the average kinetic energy $E_{k}=m\langle v^{2}(t)\rangle/2$ can be shown to
be continuous and the average potential energy $E_{p}=k(t)\langle
q^{2}(t)\rangle/2$ has the instantaneous gain:

$\displaystyle E_{k}(0^{+})=E_{k}(0^{-}),\quad
E_{p}(0^{+})=E_{p}(0^{-})+\langle W\rangle.$ (53)

Here $E_{k}(0^{-})=E_{p}(0^{-})=T/2$ and $\langle W\rangle$ is the mean work
on the system given by Eq. (7). In other words, immediately after the quench
the work is adopted by the system entirely in the form of potential energy.

One can show that for the delta protocol the situation is, in a sense,
opposite: At $t=0$ the average potential energy is continuous, whereas the
kinetic energy receives instantaneously a finite gain,

$\displaystyle E_{p}(0^{+})=E_{p}(0^{-}),\quad
E_{k}(0^{+})=E_{k}(0^{-})+\langle W\rangle.$ (54)

Here again $E_{k}(0^{-})=E_{p}(0^{-})=T/2$ and $\langle W\rangle$ is the mean
work on the system given by Eq. (14). Thus, immediately after the delta
perturbation the potential energy is unaffected, and the work on the system is
utilized as the system’s kinetic energy.

The average potential energy is determined by $\langle q^{2}(t)\rangle$.
Squaring Eq. (44) and taking the average with the equilibrium distribution
$\rho_{0}$ yields

$\displaystyle\langle
q^{2}(t)\rangle=\frac{T}{k_{0}}\left\\{S_{0}(t)-\frac{a}{m}G_{0}(t)\right\\}^{2}+\frac{T}{m}\,G_{0}^{2}(t)+\frac{1}{m^{2}}\,\langle(G_{0}*\xi)^{2}\rangle.$
(55)

The average square of the convolution in this expression can be, using the
fluctuation-dissipation relation (41), worked out to the form

$\displaystyle\langle(G_{0}*\xi)^{2}\rangle=\frac{m\,T}{\omega_{0}^{2}}\,[1-S_{0}^{2}(t)]-m\,T\,G_{0}^{2}(t).$
(56)

Then we get

$\displaystyle\langle
q^{2}(t)\rangle=\frac{T}{k_{0}}\left\\{1-\frac{2a}{m}\,S_{0}(t)\,G_{0}(t)+\left(\frac{a}{m}\right)^{2}G_{0}^{2}(t)\right\\}.$
(57)

This expression holds for $t>0$ Taking the limit $t\to 0^{+}$ and taking into
account that $G_{0}(0)=0$, see Eq. (48), one finds

$\displaystyle\langle q^{2}(0^{+})\rangle=\langle
q^{2}(0^{-})\rangle=T/k_{0}.$ (58)

This leads to the continuity of the potential energy.

Although the mean square displacement is continuous, its derivative, which is
also the two times position-velocity correlation,

$\displaystyle f(t)=\frac{d}{dt}\,\langle q^{2}(t)\rangle=2\langle
q(t)v(t)\rangle,$ (59)

is discontinuous at $t=0$. Indeed, from Eq. (57) and (48) one finds

$\displaystyle f(0^{+})=\lim_{t\to 0^{+}}\frac{d}{dt}\,\langle
q^{2}(t)\rangle=-\frac{2aT}{mk_{0}}.$ (60)

On the other hand, for $t<0$ the system is in equilibrium, so that
$f(0^{-})=0$, and

$\displaystyle f(0^{+})=f(0^{-})-\frac{2aT}{mk_{0}}.$ (61)

We referred to the discontinuity of $f(t)$ is Sec. III, evaluating the mean
work using Eq. (31).

Consider now the average kinetic energy. Squaring and averaging of Eq. (50)
yields

$\displaystyle\langle v^{2}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{T}{k_{0}}\,\left\\{\omega_{0}^{2}G_{0}(t)+\frac{a}{m}R_{0}(t)\right\\}^{2}+\frac{T}{m}\,R_{0}^{2}(t)+\frac{1}{m^{2}}\,\langle(\xi*R_{0})^{2}\rangle.$
(62)

Similar to Eq. (56), the last term can be worked out to the form

$\displaystyle\langle(R_{0}*\xi)^{2}\rangle=m\,T\,[1-R_{0}^{2}(t)-\omega_{0}^{2}\,G_{0}^{2}(t)],$
(63)

then

$\displaystyle\\!\\!\\!\langle
v^{2}(t)\rangle=\frac{T}{m}\left\\{1\\!+\\!\frac{2a}{m}\,G_{0}(t)\,R_{0}(t)\\!+\\!\left(\frac{a}{m\,\omega_{0}}\right)^{2}\\!\\!R_{0}^{2}(t)\right\\}.$
(64)

Taking the limit $t\to 0^{+}$ we find that the velocity’s second moment is
discontinuous,

$\displaystyle\langle v^{2}(0^{+})\rangle=\langle
v^{2}(0^{-})\rangle+\left(\frac{a}{m\omega_{0}}\right)^{2}\frac{T}{m},$ (65)

where $\langle v^{2}(0^{-})\rangle=T/m$. Written for the average kinetic
energy $E_{k}(t)=m\langle v^{2}(t)\rangle/2$, this discontinuity relation
takes the form Eq. (54) and leads to an observation that the mean work
$\langle W\rangle$ performed on the system is collected by the system entirely
in the form of the kinetic energy.

##  VI Method II

One may worry about the validity of the method I exploited in Sec. IV, because
it takes for granted the ad hoc ansatz (35) for the discontinuous function
$f(t)=\frac{d}{dt}\langle q^{2}(t)\rangle=2\langle q(t)\,v(t)\rangle$. Also,
solving the Langevin equation (40) using the Laplace transformation
($\mathcal{L}$), we tacitly used the relation
$\mathcal{L}\\{\delta(t)\,q(t)\\}=q(0)$. This implies the continuity of $q(t)$
at $t=0$, which is not obvious for the delta protocol.

In this section we apply another method which does not rely on the ansatz (35)
and does not assume the continuity of $q(t)$, but instead adopts a specific
representation of the delta function. Namely, we assume that the stiffness
$k(t)$ is varied according to the piecewise-constant protocol

$\displaystyle k(t)=\begin{cases}k_{0},&t<0,\\\ k_{1}=k_{0}+a/\epsilon,&0\leq
t<\epsilon,\\\ k_{0},&t\geq\epsilon,\end{cases}$ (66)

where $\epsilon,a>0$. The perturbation has the form of a rectangular impulse
of duration $\epsilon$ and amplitude $a/\epsilon$. In the limit $\epsilon\to
0$, which will be applied later, protocol (66) converges to the delta
protocol.

The protocol (66) can be viewed as a sequence of two instantaneous quenches:
$k_{0}\to k_{1}$ at $t=0^{-}$ and $k_{1}\to k_{0}$ at $t=\epsilon$.
Accordingly, the microscopic work has two contributions,

$\displaystyle W=W_{01}+W_{10},$ (67)

where, similar to Eq.(5),

$\displaystyle
W_{01}=\frac{k_{1}-k_{0}}{2}\,q^{2}(0^{-})=\frac{a}{2\epsilon}\,q^{2}(0^{-}),\quad
W_{10}=\frac{k_{0}-k_{1}}{2}\,q^{2}(\epsilon)=-\frac{a}{2\epsilon}\,q^{2}(\epsilon).$
(68)

In order to find $q^{2}(\epsilon)$, one needs to solve the Langevin equation
(40) for $t>0$ with $k(t)=k_{1}$

$\displaystyle\ddot{q}(t)=-\omega_{1}^{2}\,q(t)-\int_{0}^{t}K(t-\tau)\,\dot{q}(\tau)\,d\tau+\frac{1}{m}\xi(t),$
(69)

where $\omega_{1}^{2}=k_{1}/m$. Solving the equation with Laplace transforms,
one finds

$\displaystyle q(t)=q(0^{-})\,S_{1}(t)+v(0^{-})\,G_{1}(t)+L_{1}(t).$ (70)

Here $L_{1}(t)$ denotes the convolution

$\displaystyle
L_{1}(t)=\frac{1}{m}\,\\{G_{1}*\xi\\}=\frac{1}{m}\,\int_{0}^{t}G_{1}(t-t^{\prime})\xi(t^{\prime})\,dt^{\prime},$
(71)

and the relaxation functions $S_{1}$ and $G_{1}$ are defined as in the
previous section but now they corresponds to the stiffness $k_{1}$ and
frequency $\omega_{1}=\sqrt{k_{1}/m}$. Then for the microscopic work
$W=W_{01}+W_{10}$ we obtain

$\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{a}{2\epsilon}\,[q^{2}-q^{2}(\epsilon)]$ (72)
$\displaystyle=$
$\displaystyle\frac{a}{2\epsilon}\left\\{[1-S_{1}^{2}(\epsilon)]\,q^{2}-G_{1}^{2}(\epsilon)\,v^{2}-L_{1}^{2}(\epsilon)\right\\}-\frac{a}{\epsilon}\left\\{S_{1}(\epsilon)G_{1}(\epsilon)\,q\,v-[S_{1}(\epsilon)\,q+G_{1}(\epsilon)\,v]\,L_{1}(\epsilon)\right\\}.$

Here and below for brevity we use the notations $q=q(0^{-})$, $v=v(0^{-})$.

We are looking for the asymptotic form of the expression (72) in the limit
$\epsilon\to 0$, when the protocol (66) converges to the delta protocol.
According to Eqs. (48) and (49) (in which one needs to replace the subscript
$0$ by $1$), the initial values of the relevant functions are

$\displaystyle G_{1}(0)=0,\quad\dot{G}_{1}(0)=1,\quad\ddot{G}_{1}(0)=0$
$\displaystyle
S_{1}(0)=1,\quad\dot{S}_{1}(0)=0,\quad\ddot{S}_{1}(0)=-\omega_{1}^{2}\approx-\frac{a}{m\epsilon}$
$\displaystyle
L_{1}(0)=0,\quad\dot{L}_{1}(0)=0,\quad\ddot{L}_{1}(0)=\frac{\xi(0)}{m}.$ (73)

Then at small $t=\epsilon$ the relaxation functions have the following
asymptotic forms

$\displaystyle G_{1}(\epsilon)=\epsilon+\mathcal{O}(\epsilon^{2}),\quad
S_{1}(\epsilon)=1-\frac{a}{2m}\,\epsilon+\mathcal{O}(\epsilon^{2}),\quad
S_{1}^{2}(\epsilon)=1-\frac{a}{m}\epsilon+\mathcal{O}(\epsilon^{2}),\quad
L_{1}(\epsilon)=\mathcal{O}(\epsilon^{2}).$ (74)

Substituting into Eq. (72) yields to the leading (zero) order in $\epsilon$,
i.e. in the limit $\epsilon\to 0$, the expression

$\displaystyle W=\frac{a^{2}}{2m}\,q^{2}-a\,q\,v.$ (75)

This coincides with the result we obtained in Sec. IV with method I.

##  VII Response

The system’s response to the delta perturbation, i.e. the dynamics for $t>0$,
is more convenient to consider with method I by solving Langevin equation
(42). Let us focus on the mean square displacement $\langle q^{2}(t)\rangle$,
which determines the response of the system’s potential energy.

Let us assume that at $t<0$ the system is in thermal equilibrium with the
distribution $\rho_{0}$, Eq. (6). In Section V we obtained for $\langle
q^{2}(t)\rangle$ the result Eq. (57). Let us re-write it as

$\displaystyle\langle q^{2}(t)\rangle=\varphi(t)\,\langle
q^{2}(0^{-})\rangle,$ (76)

where $\langle q^{2}(0^{-})\rangle=T/k_{0}$ is the equilibrium mean-square
displacement (before the perturbation is applied) and the dimensionless
response function $\varphi(t)$ has the form

$\displaystyle\varphi(t)=1-2\alpha\,S_{0}(t)\,\omega_{0}G_{0}(t)+\alpha^{2}\,[\omega_{0}G_{0}(t)]^{2}.$
(77)

Since $G_{0}(0)=0$, the initial value of $\varphi(t)$ is $1$ for any memory
kernel $K(t)$ in the Langevin equation. The behavior of $\varphi(t)$ for $t>0$
depends on a particular form of the dissipation kernel $K(t)$. Below we
consider two examples for which the relaxation functions $G_{0}(t)$ and
$S_{0}(t)$ are available in closed form.

Figure 2: The response function $\varphi(t)$, defined by Eq. (76), for the
Markovian overdamped oscillator, described by the generalized Langevin
equation with the kernel $K(t)=\gamma\,\delta(t)$, for
$\beta=\gamma/\omega_{0}=5.0$ and several positive (left) and negative (right)
values of the perturbation intensity $\alpha=a/(m\omega_{0})$.

As the first example consider the kernel

$\displaystyle K(t)=\gamma\,\delta(t),\qquad\gamma^{2}-4\omega_{0}^{2}>0,$
(78)

which corresponds to the Markovian Langevin equation for the overdamped
oscillator. According to Eq. (45), the relaxation functions have the forms

$\displaystyle\omega_{0}\,G_{0}(t)=\frac{1}{\sqrt{\beta^{2}-4}}\left\\{e^{-s_{1}(\omega_{0}t)}-e^{-s_{2}(\omega_{0}t)}\right\\},\quad
S_{0}(t)=\frac{1}{\sqrt{\beta^{2}-4}}\,\left\\{\frac{1}{s_{1}}\,e^{-s_{1}(\omega_{0}t)}-\frac{1}{s_{2}}\,e^{-s_{2}(\omega_{0}t)}\right\\},$
(79)

where dimensionless parameters $\beta,s_{1},s_{2}$ are

$\displaystyle\beta=\frac{\gamma}{\omega_{0}}>2,\quad
s_{1}=\frac{1}{2}\,(\beta-\sqrt{\beta^{2}-4}),\quad
s_{2}=\frac{1}{2}\,(\beta+\sqrt{\beta^{2}-4}).$ (80)

The corresponding response function $\varphi(t)$, Eq. (77), for $\beta=5.0$ is
shown in Fig. 2 . For positive $\alpha$, the behavior of $\varphi(t)$ is
remarkably different for smaller and larger values of $\alpha$. For
$\alpha<\alpha_{0}\approx 9.6$, $\varphi(t)$ first decreases, reaches a
minimum, and then monotonically increases back to $1$. For
$\alpha>\alpha_{0}$, after reaching a minimum, $\varphi(t)$ develops a
maximum, and then monotonically decreases back to $1$. For $\alpha<0$,
$\varphi(t)$ behaves in a similar way for all values of $\alpha$: It first
increases, reaches a maximum, and then monotonically decreases back to the
initial value $1$.

Our second example is the kernel

$\displaystyle
K(t)=\frac{\omega_{*}}{2}\,\frac{J_{1}(\omega_{*}t)}{t}=\frac{\omega_{*}^{2}}{4}[J_{0}(\omega_{*}t)+J_{2}(\omega_{*}t)],$
(81)

where $J_{n}(t)$ are the Bessel functions of the first kind. The kernel has
the absolute maximum at $t=0$ and for $t>0$ it oscillates with an amplitude
decaying with time as $t^{-3/2}$. The generalized Langevin equation with
kernel (81) describes a particle attached to a semi-infinite harmonic chain,
which plays the role of the thermal bath Weiss , for the case when the mass of
the particle and the masses of chain particles are the same. The frequency
$\omega_{*}$ has the meaning of the maximal frequency of the normal modes of
the bath.

Figure 3: The response function $\varphi(t)$ given by (84), for several
positive (left) and negative (right) values of the perturbation intensity
$\alpha=a/(m\omega_{0})$.

The oscillator described by the generalized Langevin equation with kernels
like Eq. (81) was studied in Ref. Plyukhin . Here we consider only the special
case when the oscillator frequency $\omega_{0}$ and the maximal bath frequency
$\omega_{*}$ are related as

$\displaystyle\omega_{0}=\omega_{*}/\sqrt{2}.$ (82)

In that case the relaxation functions can be presented in closed form as
follow

$\displaystyle
G_{0}(t)=\frac{2}{\omega_{*}}\,J_{1}(\omega_{*}t)=\frac{\sqrt{2}}{\omega_{0}}\,J_{1}(\sqrt{2}\,\omega_{0}\,t),\quad
S_{0}(t)=J_{0}(\omega_{*}t)=J_{0}(\sqrt{2}\,\omega_{0}\,t).$ (83)

Then the response function (77) takes the form

$\displaystyle\varphi(t)=1-2\sqrt{2}\,\alpha\,J_{0}(\sqrt{2}\,\omega_{0}t)\,J_{1}(\sqrt{2}\,\omega_{0}t)+2\,\alpha^{2}J_{1}^{2}(\sqrt{2}\,\omega_{0}t).$
(84)

Fig. 3 shows that the response function has the form of oscillations with the
amplitude decreasing with time. For small values of $\alpha$ the oscillation
amplitude is small too, so that the perturbed mean square displacement
$\langle q^{2}(t)\rangle$ remains close to the equilibrium value. For larger
values of $\alpha$ the minimal value of $\varphi(t)$ is getting closer to $0$,
whereas the maximum value increases. Suppose the delta perturbation is
followed by an instantaneous quench $k_{0}\to k_{1}$. By choosing the moment
of the quench $t_{*}$ to coincide with the time of absolute maximum (minimum)
of the response function $\varphi(t)$, the mean work for the quench
$\frac{1}{2}(k_{1}-k_{0})\langle q^{2}(t_{*})\rangle$ can be strongly
increased (decreased), compared to the case when the quench disturbs the
system in equilibrium.

##  VIII Conclusion

A special property of the instantaneous quench and delta protocols is that the
microscopic work in both cases does not depend on parameters describing
stochastic dynamics, but is completely determined by equilibrium statistics of
the initial state. Intuitively, that might be expected on the ground that both
protocols have effectively zero duration. However, in contrast to the
instantaneous quench, in order to evaluate the work for the delta protocol one
does need to take stochastic dynamics into account, though only on the
asymptotically short time scale. On that scale the properties of the relevant
relaxation (correlation) functions are generic and do not depend on the
specific form of the dissipation kernel $K(t)$ and statistics of noise in the
generalized Langevin equation.

From that perspective it is clear that the approximation of overdamped
Langevin dynamics, which implies a coarse-graining of time, is inadequate to
describe the delta protocol. The solution of the overdamped Langevin equation
for the unperturbed oscillator

$\displaystyle\gamma\,\dot{q}(t)=-\omega^{2}\,q(t)+\frac{1}{m}\,\xi(t)$ (85)

is expressed in term of the exponential relaxation function
$S(t)=\exp(-\omega^{2}t/\gamma)$. The derivative of the latter at $t=0$ is
finite, whereas beyond the overdamped approximation the derivative of the
corresponding relaxation function is strictly zero, $\dot{S}(0)=0$, see Eq.
(73) and also Eqs. (79) and (83) for specific examples. The difference is
essential. Repeating the arguments of Section VI one finds that for the
overdamped case the work diverges as the width $\epsilon$ of the perturbation
impulse goes to zero, which is, of course, an unphysical result.

It is but natural to ask to what extent the idealized delta protocol can mimic
realistic spike-like perturbations with finite duration and amplitudes.
Consider, for instance, a Gaussian protocol

$\displaystyle
k(t)=k_{0}+\frac{a}{\epsilon\,\sqrt{\pi}}\,e^{-(t/\epsilon)^{2}},$ (86)

which converges to the delta protocol when $\epsilon\to 0^{+}$. In that case
solving the Langevin equation (40) analytically appears to be not feasible.
Instead, we can take into account that the Langevin equation with the kernel
(81) describes the terminal atom of a semi-infinite harmonic chain and to
study the response of this system with numerical simulation. In that way we
found that Eq. (84) describes the response reasonably well already for
$\omega_{*}\epsilon=0.1$ (underestimating local maxima of the response
function by less than ten per cent). For $\omega_{*}\epsilon<0.01$ we found
that the response function $\varphi(t)$ and the mean work $\langle W\rangle$
for the delta and Gaussian protocols are practically indistinguishable.

Consider the specific setting with $\omega_{0}=\omega_{*}/\sqrt{2}$ and
$\alpha=a/(m\omega_{0})=\sqrt{2}$. For the delta protocol the first two moment
of the dimensionless work $w=W/T$ are $\langle w\rangle=1$ and $\langle
w^{2}\rangle=5$, see Eq. (17). For the Gaussian protocol (86) the simulation
(of about $10^{4}$ runs) gives the values $\langle
w\rangle=\\{0.99,\,0.96,\,0.91\\}$ and $\langle
w^{2}\rangle=\\{4.94,\,4.68,\,4.3\\}$ for
$\omega_{*}\epsilon=\\{0.01,\,0.05,\,0.1\\}$, respectively.

In this paper we assumed that dynamics of the system for $t>0$ is governed by
the generalized Langevin equation (40). Its derivations usually assume that at
$t=0^{-}$ the bath has equilibrated in the presence of the system. An
alternative is that the bath is equilibrated separately, i.e. in the absence
of the system. In that case the Langevin equation still can be derived under
certain model assumptions Zwanzig ; Weiss , but now it has somewhat different
form

$\displaystyle
m\,\ddot{q}(t)\\!=\\!-k(t)\,q(t)\\!-\\!m\int_{0}^{t}\\!K(t-\tau)\,\dot{q}(\tau)\,d\tau+\xi(t)-m\,K(t)\,q(0),$
(87)

which involves the additional term in the right-hand side depending on $q(0)$
(often referred to as the initial slip). Using the same methods as above, one
can show that the Langevin equation (87) leads again to the result (13) for
the work.

## APPENDIX

In this Appendix we obtain the probability distribution function $f(w)$ for
the dimensionless work $w=W/T$. The relation between $f(w)$ and the moment-
generating function $M_{w}(t)$ has the form

$\displaystyle\mathcal{B}\\{f(w)\\}=\int_{-\infty}^{\infty}e^{-tw}f(w)\,dw=M_{w}(-t),$
(A1)

where $\mathcal{B}$ denotes the bilateral Laplace transform. Evaluating the
right hand-side as

$\displaystyle M_{w}(-t)=\langle e^{-tw}\rangle=\iint
e^{-t\,W(q,v)/T}\,\rho_{0}(q,v)\,dq\,dv,$ (A2)

with $\rho_{0}$ and $W$ given by Eqs. (6) and (13), one finds

$\displaystyle
M_{w}(-t)=\frac{1}{\sqrt{1+\alpha^{2}(t-t^{2})}},\quad\alpha=\frac{a}{m\omega_{0}},$
(A3)

provided the argument of the square-root is positive
$1+\alpha^{2}(t-t^{2})>0$. The latter condition can also be written as

$\displaystyle t_{1}<t<t_{2},$ (A4)

where

$\displaystyle\quad t_{1}=\frac{1-\sqrt{4/\alpha^{2}+1}}{2}<0,\quad
t_{2}=\frac{1+\sqrt{4/\alpha^{2}+1}}{2}>0.$ (A5)

From the above relations we find that the bilateral Laplace transform of
$f(w)$ is

$\displaystyle\mathcal{B}\\{f(w)\\}=\frac{1}{\sqrt{1+\alpha^{2}(t-t^{2})}}=\frac{1}{|\alpha|\,\sqrt{(t-t_{1})(t_{2}-t)}},$
(A6)

and the region of convergence of the transform has the form (A4). The inverse
transformation involves analytical continuation of the above expression to the
complex plane and the evaluation of the Bromwich integral

$\displaystyle f(w)=\frac{1}{2\pi
i}\,\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\frac{s^{sw}\,ds}{|\alpha|\,\sqrt{(s-t_{1})(t_{2}-s)}}$
(A7)

along the vertical line $Re[s]=\gamma$, lying in the region of convergence.
Eq. (A4) suggests that $t_{1}<\gamma<t_{2}$.

Figure 4: The integration contour $\Gamma$ in Eq. (A8).

For $w>0$, we close the Bromwich contour by a semi-circle to the left,
bypassing the branch cut along the part of the real axis $x<t_{1}<0$ as shown
in Fig. 4. The resulting closed contour $\Gamma$ does not enclose any
singularity, therefore

$\displaystyle I=\frac{1}{2\pi
i}\,\int\limits_{\Gamma}\frac{s^{sw}\,ds}{|\alpha|\,\sqrt{(s-t_{1})(t_{2}-s)}}=0.$
(A8)

The contributions from the large semi-circle $C_{R}$ of radius $R$ and the
small circle $C_{r}$ about $s=t_{1}$ of radius $r$ vanish in the limits
$R\to\infty$ and $r\to 0$, and the contribution from the vertical line equals
$f(w)$. Therefore $I=f(w)+I_{C}=0$, and

$\displaystyle f(w)=-I_{C}=-\frac{1}{2\pi
i}\,\int\limits_{C}\frac{s^{sw}\,ds}{|\alpha|\,\sqrt{(s-t_{1})(t_{2}-s)}},$
(A9)

where $C=C^{+}+C^{-}$ consists of the two lines just above and below the
branch cut. For $s\in C^{+}$ (above the cut), one uses the parametrization
$s=t_{1}+\rho e^{i\pi}$, which gives
$\sqrt{(s-t_{1})(t_{2}-s)}=i\sqrt{\rho(\Delta+\rho)}$, where

$\displaystyle\Delta=t_{2}-t_{1}=\sqrt{4/\alpha^{2}+1}.$ (A10)

For $s\in C^{-}$ (below the cut), $s=t_{1}+\rho e^{-i\pi}$, and
$\sqrt{(s-t_{1})(t_{2}-s)}=-i\sqrt{\rho(\Delta+\rho)}$. Taking that into
account, from Eq. (A9) we find for $w>0$

$\displaystyle
f(w)=\frac{e^{t_{1}w}}{\pi\,|\alpha|}\,\int_{0}^{\infty}\frac{e^{-\rho
w}\,d\rho}{\sqrt{\rho\,(\Delta+\rho)}}=\frac{1}{\pi\,|\alpha|}\,e^{w/2}\,K_{0}\left(\frac{\Delta}{2}\,w\right),$
(A11)

where $K_{0}(x)$ is the modified Bessel function of the second kind.

For $w<0$, the calculations are similar but we close the Bromwich contour by a
semi-circle to the right, and the branch cut is now along the part of the real
axis $x>t_{2}>0$. Choosing an appropriate branch of the square-root function
in Eq. (A7), we find for $w<0$

$\displaystyle
f(w)=\frac{e^{t_{2}w}}{\pi\,|\alpha|}\,\int_{0}^{\infty}\frac{e^{\rho
w}\,d\rho}{\sqrt{\rho\,(\Delta+\rho)}}=\frac{1}{\pi\,|\alpha|}\,e^{w/2}\,K_{0}\left(-\frac{\Delta}{2}\,w\right),$
(A12)

Combining, for arbitrary $w$ we obtain

$\displaystyle
f(w)=\frac{1}{\pi\,|\alpha|}\,e^{w/2}\,K_{0}\left(\frac{\Delta}{2}\,|w|\right),$
(A13)

which is Eq. (22) of the main text.

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