# Arrhenius law for interacting diffusive systems

Vishwajeet Kumar  The Institute of Mathematical Sciences, CIT Campus,
Taramani, Chennai 600113, India & Homi Bhabha National Institute, Training
School Complex, Anushakti Nagar, Mumbai 400094, India  Arnab Pal
<EMAIL_ADDRESS>The Institute of Mathematical Sciences, CIT Campus,
Taramani, Chennai 600113, India & Homi Bhabha National Institute, Training
School Complex, Anushakti Nagar, Mumbai 400094, India  Ohad Shpielberg
<EMAIL_ADDRESS>Department of Mathematics and Physics, University of
Haifa at Oranim, Kiryat Tivon 3600600, Israel  Haifa Research Center for
Theoretical Physics and Astrophysics, University of Haifa, Abba Khoushy Ave
199, Haifa 3498838, Israel

###### Abstract

Finding the mean time it takes for a particle to escape from a meta-stable
state due to thermal fluctuations is a fundamental problem in physics,
chemistry and biology. For weak thermal noise, the mean escape time is
captured by the Arrhenius law (AL). Despite its ubiquity in nature and wide
applicability in practical engineering, the problem is typically limited to
single particle physics. Finding a generalized form of the AL for interacting
particles has eluded solution for a century. Here, we tackle this outstanding
problem and generalize the AL to a class of interacting diffusive systems
within the framework of the macroscopic fluctuation theory. The generalized AL
is shown to conform a non-trivial yet elegant form that depends crucially on
the particle density and inter-particle interactions. We demonstrate our
results for the paradigmatic exclusion and inclusion processes to underpin the
key effects of repulsive and attractive interactions. Intriguingly, we show
how to manipulate the mean escape time using not only temperature, but also
the particle density.

###### pacs:

Valid PACS appear here

_Introduction.—_ The celebrated Arrhenius law (AL) is a cornerstone in
physics, chemistry and biology, capturing the activation time of a system from
a meta-stable state. Thermally induced activation processes are ubiquitous in
nature, e.g. in chemical reactions, protein folding, gene expressions to name
but a few. Although each variant has its unique and intriguing features, the
universality of AL with regard to the activation barrier and the surrounding
temperature nonetheless is remarkable.

Often, the AL can be manifested within the Kramer’s reaction rate theory
Hänggi  _et al._ (1990). Usually there, one is interested in the time taken by
an overdamped particle to escape from a trapping potential $U(x)$ while
coupled to a thermal bath at temperature $T$ Gardiner  _et al._ (1985). For
the particle to escape the trap, it needs a fluctuation to grant it an excess
energy $\Delta U$ – the energy difference between the bottom of the potential
to the escape point at the top. The AL states that for weak thermal
fluctuations $D_{0}=k_{B}T\ll\Delta U$, the inverse of the mean escape time is

$\displaystyle\Phi=\tau^{-1}_{0}\rm{e}^{-\Delta U/D_{0}}.$ (1)

$\tau_{0}$ provides a microscopic time scale, which may be assessed due to
arguments by Eyring Bouchet and Reygner (2016); Eyring (1935). Importantly,
$\tau_{0}$ is non-universal as it depends on the shape of $U(x)$. The wide
applicability of the AL can be attributed to the universal exponential decay
$\rm{e}^{-\Delta U/D_{0}}$. It suggests that by varying the temperature, the
experimentally accessible mean escape time allows to infer the activation
energy $\Delta U$, independent of the non-universal prefactor $\tau_{0}$. To
understand this better, imagine a diffusive process on a multidimensional,
rugged energy landscape that can imitate chemical reactions in a network or
protein conformational dynamics from unfolded to natively folded state via
misfolded state. Presence of a variety of local minima surrounded by energy
barriers $\Delta U\gg k_{B}T$ renders a natural separation of timescales in
these systems – fast fluctuations in the well followed by slow/rare
fluctuations between the wells. In other words, the enzyme fluctuates many
times within a well (typical trajectories) before leaving it (rare
trajectories). This is a key assumption behind Eq. (1) in generic activation
processes Carmeli and Nitzan (1983); Nitzan (1988); Moffitt  _et al._ (2010);
Makarov (2015).

Despite many years of study, discoveries are still being made around the AL
and exciting applications continue to be found such as activation in the
presence of viscoelastic medium Ginot  _et al._ (2022); Ferrer  _et al._
(2021) or escape dynamics of active particles Militaru  _et al._ (2021);
Woillez  _et al._ (2020), temperature dependent activation energies Peleg  _et
al._ (2012); Dyre (2006), multiple meta-stable states Militaru  _et al._
(2021) and experiments with colloids Thorneywork  _et al._ (2020); Chupeau
_et al._ (2020). Yet, one frontier that remains surprisingly less explored is
the validity of AL in many-body systems. Indeed, even for two particles with
short-range interactions, finding the activation time seems to be a formidable
challenge. See Langer (1969) which sketches out the challenges for formulating
a theory for interacting systems. In addition to the existing slow and fast
time scales, the ‘nature’ of interaction also sets another time-scale in the
problem. Consequently, the universal exponential decay in AL may no longer
hold true due to the complex interplay of different agents and their
interactions Stirnemann and Laage (2012); Pentcheva  _et al._ (2003); Junior
_et al._ (2019); Hennig  _et al._ (2015). This is illustrated in the breakdown
of the AL for active particles Woillez  _et al._ (2019), in the case of
infinite range interactions Mukamel  _et al._ (2005); Saadat  _et al._ (2023)
as well as in low temperature glassy dynamics Dyre (2006).

Figure 1: The generalized AL for interacting diffusive systems. The minimal
energy configuration for the SSEP and SIP is sketched in the sub-figures of
(a) and (b) correspondingly. For the SSEP, the particles pack tightly, leading
to $U_{top}=U(\overline{\rho}_{0})$. For the SIP, all the particle can
accumulate on the potential minimum at $x=0$, leading to $U_{top}=0$. The $g$
function is tested numerically, as a function of $\overline{\rho}_{0}$, for
the SSEP (a) and the SIP (b) with three potentials $U(x)$: linear $\Delta Ux$,
harmonic $\Delta Ux^{2}$ and cosine $\frac{1}{2}\Delta U(1-\cos(\pi x))$. We
confirm numerically, for the SSEP $g=1-U(\overline{\rho}_{0})/\Delta U$ (left
panel) and for the SIP $g=1$ (right panel), as predicted by Eq. (2). Both
predictions are well fitted up to the expected numerical errors. We refer to
SI  for more details.

In this letter, our aim is to delve deeper into the AL for interacting
systems. However, before doing that, it is insightful to derive the AL for $M$
non-interacting particles. Notably, the AL can be computed from the large time
survival probability which can be derived by mapping the problem to an
effective Schrödinger equation with absorbing boundaries – see the textbooks
Gardiner  _et al._ (1985); Risken and Risken (1996) for this standard
procedure. Technically, the survival probability
$\mathcal{S}(t)\asymp\exp\left[-t\,\Phi\right]$ Gardiner  _et al._ (1985),
111Assuming $t\gg L^{2}/D_{0}$ the diffusion time. For the short time limit,
see Agranov and Meerson (2018)., where $\Phi=D_{0}M\Lambda$ and $\Lambda$ is
the ground state energy of the Fokker-Planck Hamiltonian
$\hat{H}_{FP}=-D_{0}\partial_{xx}+\frac{D_{0}}{4}(\partial_{x}U/D_{0})^{2}-\frac{1}{2}\partial_{xx}U$
SI ; Sabhapandit and Majumdar (2020). Evidently, the AL universality still
prevails for many non-interacting particles. What happens to this universality
when we incorporate many-body interactions? Here, we try to address this
important question.

Consider an extended 1D system of interacting diffusing particles in a
monotonically increasing potential $U(x)$ with initial mean density
$\overline{\rho}_{0}=M/L$, where $M$ is the number of particles and $L$ is the
domain size. We show, in this letter, that the mean time it takes for a
particle to escape from the potential is given by the following generalized AL

$\Phi\asymp\rm{e}^{-\Delta Ug(\overline{\rho}_{0})/D_{0}},$ (2)

where $g$ depends both on the density $\overline{\rho}_{0}$ and on the inter-
particle interactions. By minimizing the particle configuration energy, one
can show $g=1-U_{top}/\Delta U$ where $U_{top}$ is the highest energy a
particle can attain in the minimum energy configuration. The salient point
here is that the exponential form of the AL is still preserved. However, the
generalized AL crucially depends on the exact nature of the inter-particle
interactions as well the density.

To illustrate the generalized AL, consider the paradigm of simple symmetric
exclusion process (SSEP) where the inter-particle interaction is a hard-core
exclusion Mallick (2015). Minimizing the particle configuration energy implies
that they are tightly packed around the potential minimum, resulting in $g<1$
[see Fig. 1(a), details later]. Therefore, the repulsive interaction imposes a
shorter mean escape time by (2). Eq. (2) is the central result of this letter
and we will prove it for a class of diffusive interacting systems, within the
framework of the macroscopic fluctuation theory (MFT) Bertini  _et al._
(2015).

_Macroscopic Fluctuation Theory.—_ Over the last two decades, the Macroscopic
Fluctuation Theory (MFT) has been instrumental to understand nonequilibrium
fluctuations in diffusive systems Derrida  _et al._ (2001); Bertini  _et al._
(2002); Appert-Rolland  _et al._ (2008); Shpielberg  _et al._ (2018); Tailleur
_et al._ (2008); Derrida and Sadhu (2019a); Hurtado  _et al._ (2014); Mallick
_et al._ (2022); Derrida (2007); Bertini  _et al._ (2006); Imparato  _et al._
(2009); Baek  _et al._ (2017); Bettelheim  _et al._ (2022); Lazarescu (2015);
Tizón-Escamilla  _et al._ (2017); Agranov  _et al._ (2023). Quite recently,
the MFT was also used to capture the survival probability of interacting
diffusive particles from a domain Agranov and Meerson (2018). Here, we extend
this formalism in the presence of a potential which naturally allows us to
compute the survival probability and study the generalized AL.

To set the stage, let us consider a $1D$ system of size $L$. The system is
occupied with interacting diffusive particles that satisfy the continuity
equation $\partial_{s}\rho=-\partial_{x}j$ with the density and current
density $\rho(x,s)$ and $j(x,s)$ respectively. Here, $x\in\left[0,1\right]$
and $s\in\left[0,t\right]$ are diffusively rescaled Derrida (2007). The
fundamental formula of the MFT asserts that the path probability is

$\displaystyle\text{Prob}\left[\\{\rho,j\\}\right]$ $\displaystyle\asymp$
$\displaystyle\exp{\left(-\frac{L}{4D_{0}}\intop^{1}_{0}dx\intop^{t}_{0}ds\,\frac{(j-J(\rho))^{2}}{\chi(\rho)}\right)}$
$\displaystyle J(\rho)$ $\displaystyle=$
$\displaystyle-D(\rho)\partial_{x}\rho-\chi(\rho)\partial_{x}\tilde{U},$ (3)

where we have defined $\tilde{U}=U/D_{0}$ as the rescaled potential. $D(\rho)$
and $\chi(\rho)$ are the density dependent diffusivity and mobility which
encapsulate the diffusive dynamics. In Eq. (Arrhenius law for interacting
diffusive systems), the continuity equation is implicitly assumed.

Moreover, we assume that the particles are constrained between one reflecting
and one absorbing wall at $x=0$ and $x=1$. Thus, $\mathcal{S}(t)$, the
survival probability of the particles to stay inside the region upto time $t$
can be written as a conditional sum over all the paths
Prob$\left[\\{\rho,j\\}\right]$ that satisfy the following boundary conditions

$J(\rho)|_{x=0}=\rho_{x=1}=0,$ (4)

and the mass conservation in the system $L\int dx\rho(x,t)=M$.

Within the MFT, one would expect $\mathcal{S}(t)\asymp\rm{e}^{-t\,\Phi}$,
where computation of $\Phi$ reduces to a minimization problem of finding an
optimal fluctuation $\\{\rho,j\\}$ that satisfy the above mentioned
constraints. Note that the optimal fluctuation governs $\Phi$ due to the large
$L$ saddle dominated probability in (Arrhenius law for interacting diffusive
systems), however the minimization problem still remains hard. To address
that, we use the additivity principle (AP), that was introduced in Bodineau
and Derrida (2004) and proved as a useful tool in evaluating large deviations
Bertini  _et al._ (2015) (also see Shpielberg and Akkermans (2016); Shpielberg
_et al._ (2017, 2018); Appert-Rolland  _et al._ (2008)). The AP posits that
the optimal fluctuation density is time-independent, reducing the complexity
of finding the optimal fluctuations.

For a $1D$ system, the AP assumption $\rho(x,t)=\rho(x)$ implies
$j(x,t)=$const due to the continuity equation. However, const$=0$ since the
current vanishes at the boundaries. Finding $\Phi$ then reduces to the
following minimization problem

$\Phi=\frac{D_{0}L}{4}\min_{\rho(x)}\int
dx\,\mathcal{L}(\rho,\partial_{x}\rho),~{}\mathcal{L}=\frac{J(\rho)^{2}}{\chi}-4\Lambda(\rho-\overline{\rho}_{0}),$
(5)

where $\rho(x)$ is subjected to (4) and $\Lambda$ is Lagrange multiplier
ensuring the mass conservation $\overline{\rho}_{0}=\int dx\rho(x)$.

The minimization problem in (5) then boils down to solving an Euler-Lagrange
(EL) equation with the Lagrangian $\mathcal{L}$. Using the transformation
$F(\rho)=\int^{\rho}_{0}dz\frac{D(z)}{2\sqrt{\chi(z)}}$ (see Agranov and
Meerson (2018)), the resulting EL reads

$-\partial_{xx}F+\frac{1}{8}(\partial_{x}\tilde{U})^{2}\chi^{\prime}-\frac{1}{2}\sqrt{\chi}\partial_{xx}\tilde{U}=\Lambda\frac{\sqrt{\chi}}{D},$
(6)

where $\chi=\chi(\rho[F])$ and $\chi^{\prime}=\delta\chi/\delta F$. The
resulting boundary conditions inherited from (4) are
$J_{F}|_{x=0}=\rho[F]|_{x=1}=0$ with rescaled current
$J_{F}=\partial_{x}F+\frac{1}{2}\chi^{1/2}\partial_{x}\tilde{U}$. Thus, the
survival probability can be estimated via (5) as

$\Phi=D_{0}L\int dxJ^{2}_{F},$ (7)

where $F$ is the solution of (6). Before discussing the interactions, it is
instructive to re-derive the results for non-interacting particles. Indeed,
here $D=1,\chi=\rho$ so that $\rho=F^{2}$ and (6) recovers
$\hat{H}_{FP}F=\Lambda F$, which is essentially the eigen-value problem for
the non-interacting system. Skipping details, one can show that
$\Phi=D_{0}L\overline{\rho}_{0}\Lambda$ SI  which implies that $\Lambda$ is
the ground state energy of $\hat{H}_{FP}$ as found earlier.

Furthermore, assuming short range interactions, the limit of a dilute system,
i.e. $\overline{\rho}_{0}\rightarrow 0$, suggests interactions become
negligible. Indeed in this limit, $F\rightarrow 0$. Thus, Eq. (6) can be
linearized in $F$, leading to the non-interacting case. Thus for dilute
systems, the AL is recovered. In what follows, in order to justify the
generalized AL in Eq. (2), we turn to study two processes of interacting
systems: the SSEP for repulsive interactions and the symmetric inclusion
process (SIP) for attractive interactions.

_Repulsive interaction.—_ The SSEP is a lattice gas model where each lattice
site $x$ has an occupancy $n_{x}=\\{0,1\\}$. In the SSEP, a particle is
allowed to hop to a nearest neighbour with a fixed rate, provided the target
site is empty. Despite its simplicity, the SSEP has been widely studied in the
context of nonequilibrium statistical mechanics and has been used as a
paradigmatic model for understanding the behavior of interacting particle
systems in many physical and biological systems, e.g. traffic flow, protein
synthesis, and gene regulation Mallick (2015). The SSEP typically demonstrates
genuine nonequilibrium behavior, e.g. a non-product steady state measure and
long range nonequilibrium correlations Bertini  _et al._ (2015); Derrida
(2007). Furthermore, the SSEP is susceptible to exact solutions;
microscopically via the Bethe ansatz and macroscopically through the MFT
Appert-Rolland  _et al._ (2008); Shpielberg  _et al._ (2018); Derrida and
Sadhu (2019b, a); Mallick  _et al._ (2022).

For the SSEP, $D=1,\chi=\rho(1-\rho)$ Derrida (2007). The transformation to
$F$ leads to $\rho=\sin^{2}F$. We restrict $F$ to the range
$\left[0,\pi/2\right]$ to ensure a positive $\chi^{1/2}=\frac{1}{2}\sin 2F$.
While $\chi,\chi^{\prime}$ in (6) becomes explicit, a solution for arbitrary
potential is challenging. Fortunately, an analytical solution can be obtained
for the linear potential $\tilde{U}(x)=\Delta\tilde{U}x$. In that case (6) is
simply an autonomous equation. With the transformation $y=\cos 2F$, and by
assuming that the density as well as $y$ are monotonous functions, (6) can be
reduced to a first order differential equation SI

$\displaystyle\frac{dy}{dx}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Delta\tilde{U}\sqrt{1-y^{2}}\sqrt{1-y^{2}+8\lambda(y-y_{0})},$
(8) $\displaystyle y_{0}=y|_{x=0},\quad y_{1}=y|_{x=1}=1,$

where $\lambda=\Lambda/\Delta\tilde{U}^{2}$. Note that $-1\leq y_{0}\leq 1$.
Here $y_{0}\rightarrow 1(-1)$ implies that the density at the reflecting
boundary is approaching zero (unity).

Eqs. (7) and (8) constitute an implicit solution of the survival probability,
denoted by $\Phi_{\rm SSEP}$, for the SSEP. However, it is worth stressing
that one need not explicitly solve $y$ (and thus $F$) in order to obtain
$\Phi_{\rm SSEP}$. Skipping details from SI , we obtain

$\displaystyle\Phi_{\rm SSEP}$
$\displaystyle=\frac{C_{0}}{2}\left(y_{0}-1+C_{0}-C_{2}+4\lambda(C_{1}-y_{0}C_{0})\right),$
$\displaystyle\text{where~{}~{}}C_{k}$
$\displaystyle=\int^{1}_{y_{0}}dy\frac{y^{k}}{\sqrt{1-y^{2}}\sqrt{1-y^{2}+8\lambda(y-y_{0})}},$
(9)

for $k=0,1,2$. Fortunately, the $C_{k}$ integrals are analytical and involve
elliptic functions. Furthermore, $\Phi_{\text{SSEP}}$ is given in terms of
($\lambda,y_{0}$) which are implicit functions of $\Delta\tilde{U}$ and
$\overline{\rho}_{0}$.

To make the relations explicit, we notice that direct integration of (8) leads
to $C_{0}=\Delta\tilde{U}/2$. Also, recalling that $\overline{\rho}_{0}=\int
dx\sin^{2}F$ implies that $2\overline{\rho}_{0}=1-C_{1}/C_{0}$. The regime of
large $\Delta\tilde{U}$ is supported by $0<1+y_{0}\ll\lambda\ll 1$. In these
limits and to leading order
$\lambda=2\rm{e}^{-\Delta\tilde{U}(1-\overline{\rho}_{0})}$ and
$(1+y_{0})=\rm{e}^{-\Delta\tilde{U}\overline{\rho}_{0}}$. Finally, we have

$\displaystyle\Phi_{\rm
SSEP}=D_{0}LA\rm{e}^{-\Delta\tilde{U}(1-\overline{\rho}_{0})},$ (10)

where $A$ is a polynomial function of $\overline{\rho}_{0}$ and
$\Delta\tilde{U}$ SI .

At this point, we connect the above with our announced result (2). Eq. (10)
implies $g=1-\overline{\rho}_{0}$ for the linear potential. In the exclusion
process, the system’s energy is minimized if the particles are ordered as
close to the potential minimum as possible. Consequently, the minimum energy
occurs when the system is fully occupied between $x=0$ and
$x=\overline{\rho}_{0}$, and therefore $U_{top}=U(x=\overline{\rho}_{0})$ (see
Fig. 1).

Although solving Eq. (6) is hard for generic monotonous potentials, the same
physical arguments can be given to infer the generalized AL. Indeed, the
minimum energy attained by tight packing the exclusion particles in monotonous
potentials, lead to $U_{top}=U(x=\overline{\rho}_{0})$. To demonstrate this
behavior, we employ a standard shooting method algorithm to solve the eigen-
problem (6) for various potentials (see SI  for more details). The numerical
results provide an excellent agreement with Eq. (2) thus validating the
generalized AL for arbitrary monotonous potentials.

Attractive interaction.—We now turn our attention to the Symmetric Inclusion
Process (SIP) where the interaction among the particles is attractive. In the
SIP, the jump rate of a particle from site $x$ with occupancy $n_{x}$ to a
neighboring site $y$ with occupancy $n_{y}$ is $n_{x}(1+n_{y})$. Thus,
multiple particles can occupy a given site and the dynamics makes particle
clustering favorable Grosskinsky  _et al._ (2011, 2013); Reuveni  _et al._
(2012).

For the SIP, one finds $D=1,\chi=\rho(1+\rho)$. The $F$-transformation implies
$\rho=\sinh^{2}F$. Notice that unlike the SSEP, $F$ is unbounded for the SIP.
Similarly to the SSEP analysis, one can explicitly write the Euler-Lagrange
equation (6), but an explicit solution is again challenging for an arbitrary
potential. However, some analytical progress can be made for the linear
potential which we discuss in the following.

In this case, Eq. (6) becomes an autonomous equation just like for the SSEP.
Using the transformation $y=\cosh 2F$ and assuming the the monotonicity of $F$
leads us to SI

$\frac{dy}{dx}=-\frac{1}{2}\Delta\tilde{U}\sqrt{y^{2}-1}\sqrt{y^{2}-1+8\lambda(y_{0}-y)},$
(11)

where we note that $1\leq y\leq y_{0}$ and $y_{0}=y|_{x=0}$ may be unbounded.
Similarly, to the SSEP, one can express the survival probability
$\Phi_{\text{SIP}}$ for the SIP as SI

$\Phi_{\text{SIP}}=\frac{C_{0}}{2}\left(1-y_{0}+C_{2}-C_{0}+4\lambda(y_{0}C_{0}-C_{1})\right).$
(12)

Here, we have re-defined the $C_{k}$ integrals for the SIP

$C_{k}=\int^{y_{0}}_{1}\frac{y^{k}dy}{\sqrt{y^{2}-1}\sqrt{y^{2}-1+8\lambda(y_{0}-y)}},$
(13)

which are analytic. Integrating (11) over the whole range implies
$C_{0}=\Delta\tilde{U}/2$. Then, using $\overline{\rho}_{0}=\int dx\sinh^{2}F$
we find $C_{1}-C_{0}=\Delta\tilde{U}\overline{\rho}_{0}$. The large
$\Delta\tilde{U}$ regime occurs for $\lambda\ll\mu=y_{0}\lambda\ll 1$. Then,
following SI  we find
$\lambda=\rm{e}^{-\Delta\tilde{U}(1+\overline{\rho}_{0})}$ and
$\mu=2\rm{e}^{-\Delta\tilde{U}}$ to leading order. This results in

$\displaystyle\Phi_{\text{SIP}}=D_{0}LA\rm{e}^{-\Delta\tilde{U}},$ (14)

where $A$ once again is a polynomial function of
$\overline{\rho}_{0},\Delta\tilde{U}$ SI . In the non-interacting limit
$\overline{\rho}_{0}\rightarrow 0$ with comparable $\mu$ and $\lambda$ and
thus, we recover the AL $\Phi_{\text{SIP}}\propto\rm{e}^{-\Delta\tilde{U}}$ SI
.

Eq. (14) remarkably coincides with the single particle AL. This indeed stems
from the underlying physics of the SIP, where the attractive interactions
imply that the minimum energy configuration is attained when all the particles
bunch at the potential minimum. Comparing $\Phi_{\text{SIP}}$ with Eq. (2)
results in $g=1$. In fact, this holds for any monotonous potential. To
demonstrate this fact, we employ our shooting algorithm to solve (6) for
different potentials. The numerical results are illustrated in Fig. 1
validating the AL for SIP regardless of the particle density
$\overline{\rho}_{0}$. We should stress that the simplicity of Eq. (14) can be
deceptive since there is no simple or trivial way to derive it.

Discussion.— Understanding escape times in a thermal activation process while
accounting for many-body interactions is at the heart of this letter. To this
end, we have studied the survival probability of interacting particles with
diffusive dynamics in a potential trap. In the limit of weak thermal
fluctuations $\Delta U\gg D_{0}$, we show, using the seminal macroscopic
fluctuation theory, that the Arrhenius law can be generalized (Eq. (2)) with
$\Delta Ug$ – an interaction-dependent effective activation energy. We
demonstrate our result for two distinct interaction classes. Particles which
repel themselves reveal a decrease in $g$ leading to a facilitated escape from
the potential. On the other hand, for purely attractive particles barring
condensation, we reach quite surprisingly and non-trivially to an effective
single particle problem. It should be stressed that albeit its technical
simplicity and intuitively appealing physical interpretations, any attempts to
generalize the insight gained from single particle AL to many particles with
arbitrary interactions remains exorbitantly challenging. Furthermore, the
generalized AL holds in store quite a few surprises.

Naively, one expects that the generalized AL would be more restrictive from an
experimental stand point. However, Eq. (2) suggests the opposite with two
immediate interesting directions. First, consider e.g., single file colloids
in a potential trap, represented by exclusion particles. By controlling the
particle density in the trap, one can infer from $\Phi$ the exact shape of the
potential Bryan  _et al._ (2022); Dudko  _et al._ (2006); Gieseler  _et al._
(2021). Second, experimental control of the potential affecting the colloids
as in Thorneywork  _et al._ (2020) together with Eq. (2), allows to infer the
particle density in the system.

Beyond these applications, this work opens new avenues of research. First, it
will be important to extend Eq. (2) to include arbitrary potential landscape,
possibly also to higher dimensions Pérez-Espigares  _et al._ (2016); Akkermans
_et al._ (2013). Additionally, it is interesting to consider interactions that
cannot be classified as purely attractive/repulsive e.g., the Katz-Leibowitz-
Spohn models Katz  _et al._ (1983); Shpielberg (2017). A yet unexplored path
is to consider the AL for multi-species particles, where $D,\chi$ become
matrices Bodineau and Derrida (2011). Exploring how inter-species interactions
can serve to speed up/ slow down the activation time would be illuminating and
potentially useful both in theory and experiments involving transport of
particles or ions through channels.

_Acknowledgements.—_ A.P. gratefully acknowledges research support from the
Department of Science and Technology, India, SERB Start-up Research Grant
Number SRG/2022/000080 and the Department of Atomic Energy, India. O.S. thanks
Tridib Sadhu and Baruch Meerson for fruitful discussions. O.S. also
acknowledges the support of the Erwin Schrödinger International Institute for
Mathematics and Physics during the thematic program DPS22, where the
discussions took place.

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# A first look into Utiq: Next-generation cookies at the ISP level

1st Ismael Castell-Uroz  Universitat Politècnica de Catalunya
Barcelona, Spain
<EMAIL_ADDRESS>2nd Pere Barlet-Ros  Universitat Politècnica de
Catalunya
Barcelona, Spain
<EMAIL_ADDRESS>
###### Abstract

Online privacy has become increasingly important in recent years. While third-
party cookies have been widely used for years, they have also been criticized
for their potential impact on user privacy. They can be used by advertisers to
track users across multiple sites, allowing them to build detailed profiles of
their behavior and interests. However, nowadays, many browsers allow users to
block third-party cookies, which limits their usefulness for advertisers. In
this paper, we take a first look at Utiq, a new way of user tracking performed
directly by the ISP, to substitute the third-party cookies used until now. We
study the main properties of this new identification methodology and their
adoption on the 10K most popular websites. Our results show that, although
still marginal due to the restrictions imposed by the system, between 0.7% and
1.2% of websites already include Utiq as one of their user identification
methods.

###### Index Terms:

web tracking, cookie, Utiq

##  1 Introduction

Online privacy is an increasingly important concern in today’s digital age, as
more and more of our personal information is stored and transmitted through
the internet. One of the main concerns is the amount of data that companies
collect about users. From search engines to social media platforms, companies
gather vast amounts of information about browsing habits, purchases, and
interests. Until now, third-party cookies have been the most commonly used
solution to perform this kind of identification. These cookies allow websites
to track the user’s activity across multiple sites, collecting information
about their interests and behavior. This information is often used to
personalize advertising content and improve marketing strategies, which is why
many companies rely on it for their business operations. However, as concerns
over online privacy have grown, many browsers now block third-party cookies by
default (e.g., [1, 2]), making them less useful for tracking user activity
across the web.

In this context, Deutsche Telekom, Movistar, Vodafone, and Orange, four of the
biggest telecommunications companies in Europe, have recently developed Utiq,
their own identification system with a special focus on privacy and
transparency, to replace third-party cookies. Utiq works by identifying the
Internet connection tied to the user and generating from it a pair of
anonymized tokens that can be used by companies for marketing and advertising
purposes. In this paper, we explore the main properties of Utiq and evaluate
the level of adoption of this new system after six months of its inception. To
the best of our knowledge, no previous work has studied this new
identification methodology. Moreover, we also study if websites using Utiq
also use other web tracking methods that, combined with Utiq’s identification
tokens, may be used to improve their user profiling systems, which would
render the new system as controversial as third-party cookies. Our results
show that, although still marginal, six months after the creation of the
service and depending on the country, between 0.7% and 1.2% of the 10k most
popular websites include Utiq as one of their identifying processes. However,
we also demonstrate that all of them complement this identification system
with more intrusive methods, such as canvas or font fingerprinting [3], not
resulting in a real privacy improvement over the third-party cookies that Utiq
substitutes.

The rest of the paper is organized as follows: Section 2 reviews the most
relevant related work. Section 3 describes the fundamentals of Utiq. Section 4
presents the study of its level of adoption, and Section 5 discusses some
considerations about tracking as well as some of the challenges that Utiq must
face in order to get a wider adoption. Lastly, Section 6 concludes the paper.

##  2 Related work

Web tracking has become an increasingly significant issue in recent years due
to its potential impact on user privacy. One of the most prevalent web
tracking methods has been third-party cookies. These files can be used to
identify users across multiple websites. While they have been widely used for
years, they have also been criticized for their potential impact on user
privacy as they allow companies to build detailed profiles of their users
behavior and interests. As such, there are many works in the literature
studying the cookie ecosystem and its implications. In [4], Leon et al. were
one of the first to study the impact on usability of different tools that
block cookies to improve the privacy of the user. Englehard et al. study in
[5] how third-party cookies can be used to link visits from the same user to
different websites even if the user’s IP address is different. In [6],
Papadopoulos et al. study the proliferance of cookie syncing, a technique that
allows different and not related third-party companies to sync the information
about their users cookies in order to identify them even in services that are
not self-owned.

In response to concerns about web tracking and user privacy, many websites
have implemented privacy policies that outline how they collect and use data
about their users. While privacy policies can help users understand what data
is being collected about them, they are not always sufficient to ensure user
privacy. To address these issues, some countries have implemented legal
frameworks aimed at protecting user privacy online. For example, the European
Union’s General Data Protection Regulation (GDPR [7]) requires websites to
obtain explicit consent from users before collecting certain types of data.
However, these laws are not always effective at preventing all forms of web
tracking. Thus, many browsers have followed a more simple but effective way of
limiting this tracking behavior: blocking all third-party cookies by default
on all websites (e.g., Firefox [1], Safari [2]). Even Google Chrome, the most
widely used browser, is currently progressively introducing this measure [8].

However, most online companies rely on this kind of information for their
business models. Thus, many of them are choosing to use more intruding
techniques, such as browser or device fingerprinting, to obtain personal data.
These techniques identify the device being used to browse the web by obtaining
as much information as possible about it in order to create identifying
fingerprints. We refer the interested reader to [3] for a comprehensive survey
with information about this kind of web tracking mechanism. On the other hand,
works like [9] from Nikiforakis et al. or [10] from Englehardt et al. have
found a rise in the use of this kind of technique during the last years.
Castell-Uroz et al. in [11] quantify in more than 95% of the total the number
of websites that use fingerprinting or similar transparent tracking methods
online. Although the research community has presented many works to
automatically find and block this kind of web tracking by multiple means
(e.g., [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]), the difficulty of
adoption of those techniques makes this still an open problem.

In this paper, we examine Utiq, a different approach taken by the some of the
most important Internet service providers in Europe to substitute the use of
third-party cookies. Utiq is based on the identification of the user by means
of a set of anonymized tokens generated directly from the information about
the connection of the user.

TABLE I: Utiq compliant ISPs

Country ISP France Orange Bouygues Telecom SFR Germany Deutsche Telekom
Vodafone Congstar Fraenk Spain Movistar Orange Jazztel Simyo

##  3 Utiq

Until recently, online companies relied heavily on third-party cookies for
marketing and advertisement purposes. However, many browsers now block these
cookies by default (e.g., Firefox [1], Safari [2]), with Google Chrome
expected to do so in 2024 [8]. In response, Utiq [23], a joint venture between
Deutsche Telekom, Movistar, Vodafone, and Orange, four of the major European
telecommunication operators, has developed its own digital user identification
platform as an alternative to third-party cookies. The platform reportedly
focuses on privacy and transparency, always asking for users consent before
their identification. While currently it is only available in Germany, Spain,
and France, the company plans to expand its services to the United Kingdom and
Italy in 2024. Table 1 provides a comprehensive list of internet service
providers that allow user identification through Utiq’s technology, including
the main companies and their affiliates in these three countries.

Figure 1: Utiq diagram

Utiq functions by acquiring what they refer to as a network signal called
consentpass. The consentpass is an identifier based on the user’s Internet
connection details and tied to the real user behind it. It is then used to
generate two anonymized and distinct signals, the martechpass and adtechpass.
These signals serve different purposes within the advertising ecosystem. The
martechpass focuses on marketing objectives and is generated individually for
each brand or publisher that incorporates Utiq services. The marketing
ecosystem seeks to identify users in order to deliver personalized messaging
to engage with customers (e.g., social media, email, or interactive content
marketing). From Utiq’s perspective, a single publisher can share the
martechpass between multiple services as long as the user explicitly consents
to sharing this kind of information. In contrast, the adtechpass is employed
for programmatic advertising within the adtech ecosystem. The advertising
ecosystem aims to find the most effective way to promote a product or service
through paid or ”sponsored” channels. The adtechpass is used to uniquely
identify users and helps to prevent inefficiencies such as displaying the same
advertisement repeatedly. This allows advertisers to optimize their
expenditures.

Utiq offers a single location called Consenthub for managing all online
service permissions. Within this platform, users can fully deactivate Utiq
service usage for up to a year. Although websites may continue requesting user
consent to utilize Utiq services during this timeframe, the identification
process will not proceed. Afterward, the status resets, and users can either
approve specific service consents or temporarily disable Utiq again for
another year. As Utiq is a system aimed at being deployed in European
countries, Consenthub complies with all the regulations introduced in the GDPR
[7].

Fig. 1 showcases Utiq’s identification methodology. Initially, to retrieve the
network signal and generate the two identifying tokens, users must first
accept all privacy policies on the main website. The website then downloads a
JavaScript file (utiqLoader.js) responsible for checking that all the required
privacy consent conditions are met by the user. This file is provided through
the ”utiq” subdomain of the primary website (e.g.,
utiq.example.com/utiqLoader.js). This subdomain must be redirected via a CNAME
record to the URL frontend.prod.utiq-aws.net, where the latest version of the
file can be accessed. Once downloaded and executed, the script checks for two
first-party cookies called “utiq_consent_status” and “connectId.” If the first
cookie is not available, a new consent manager is loaded to obtain user
consent for generating and using Utiq identifiers. The second cookie only
exists if the user has already given consent to load it and identifies the
network signal used to generate the consentpass. This cookie mainly serves to
decrease the burden on ISPs of checking connection details every time. Both
cookies are valid for 90 days, after which the user must explicitly give
consent again through Utiq’s consent manager. Once the connection details are
known, the JavaScript checks the connection by means of a call to
“utiq.example.com/op/idconnect/mno-precheck” and saves it inside a new object
in the browser’s local storage called utiqEligibility. This object identifies
if the ISP being used to open the website is eligible for using the service or
not, or, in other words, if it is one of the ISPs present in table I. It also
verifies if the specific website being opened has already been revoked from
Utiq’s Consenthub or if the Utiq service has been stopped altogether. In both
cases, it will halt the loading process of the two identification tokens. Only
when the user has given explicit consent to both the main online service and
Utiq’s service and the connection is done through one of the compliant ISPs
will both tokens be available through a new object in the browser’s local
storage called utiqPass.

Finally, Utiq also provides a way to programmatically integrate Utiq’s
adtechpass to an SSP (Supply-Side Platform) by means of the prebid.js library.
SSPs are interfaces that publishers use to automatically sell their remnant
(ad space not sold through direct deals) inventory. The prebid.js library
includes not only Utiq but also many other user identification services such
as Merkle, Criteo, or Quantcast. Regarding the programmatic connection to the
DSP (Demand-Side Platform), Utiq has an exclusivity agreement with AdForm
[24], one of the most important DSPs. Thus, websites that use Utiq to
automatically sell their advertisement space will only receive advertisements
included in the portfolio of AdForm.

##  4 Adoption study

Utiq started operating in August 2023. With most browsers now blocking third-
party cookies by default and Google’s Chrome progressively introducing the
same measure during 2024, this period should be decisive to determine the
acceptance that this new system will have during the next few years. In this
section, we study the current adoption of the system, almost six months after
the starting date.

###  4.1 Methodology

Utiq is currently available only in three countries: France, Germany, and
Spain. However, we studied the top 100k most popular websites from the Tranco
List111Available at https://tranco-list.eu/list/24W99 [25] generated on
January 15th 2024, regardless the country where it is hosted, to discover
possible foreign domains identifying European users. For each website in our
dataset, we tried to directly load the URL
”https://utiq.domain.com/utiqLoader.js”, one of the requirements websites have
to follow in order to use Utiq for their identification systems. The
inspection was directly done by means of a combination of Python and the CURL
tool. Moreover, the URL ”https://utiq.domain.com/op/idconnect/mno-precheck”
was also inspected. This address should return a {“status”:“ok”} message if
the ISP supports Utiq or {“status”:“Not found”} otherwise. Only in the event
that both URLs are accessible will we account for the domain as compliant with
Utiq, as per their requirements.

TABLE II: Utiq presence

Country | Websites | Utiq compliant | Percentage
---|---|---|---
Total | 9316 | 83 | 0.89%
France | 2048 | 14 | 0.68%
Germany | 3591 | 27 | 0.75%
Spain | 3641 | 42 | 1.15%

###  4.2 Results

Fortunately, all the domains using Utiq pertained to one of the three
countries where Utiq is available. Thus, other countries, especially non-GDPR
compliant ones, are not using the system yet. To inspect the adoption rate we
selected the subset of domains pertaining to one of those three countries
(about 10k websites) and checked Utiq prevalence for each of them. Table II
presents the overall obtained results for the subset of all those domains as
well as those divided by country. Although still quite marginal, there are
differences in adoption between the three countries, with Spain leading with a
rough 1.2%, while Germany and especially France show less introduction with
0.75% and 0.68%, respectively. Interestingly, all the websites included in the
list except three of them are online newspapers. From the other three
websites, two are websites about cooking recipes, and the last one is “atf-
tagmanager.de”, a web tracking service that seems to have included Utiq’s
identifier in their identification processes. One website, “actu.fr”, loaded
the utiqLoader.js file before getting the consent of the user in some of our
visits. This goes against Utiq requirements and is possibly a configuration
error. Finally, about 80% of the websites including Utiq (66 out of 83)
integrate programmatic selling processes via the prebid.js library. Appendix A
contains the complete list of websites compatible with Utiq services, as well
as their country and their rank within the Tranco list.

Although Utiq emphasizes their private and transparent identification methods,
the process is similar to that of actual third-party cookies, with a unique
token generated per user available for the internal website identification
systems. In fact, comparing both approaches, Utiq may be considered more
intrusive as the tokens are based on completely unique parameters that can
precisely identify the user through time, which is impossible with regular
cookies that the user can periodically clean. Moreover, websites including
Utiq services can also use other web tracking systems to complement their
identification processes to better profile the user, and here Utiq can
represent an important concern by following users over time.

Figure 2: Complementary tracking methods

To study this situation, we explored other tracking methods that Utiq-
compliant websites may also be using. This information was collected using ORM
[11], a tool to perform studies about web tracking on a given population of
websites. We inspected not only the homepage but also all the links contained
in it that pertain to the same domain. Figure 2 presents the distribution of
other tracking methods currently included in the 83 websites detected using
Utiq. As shown, all the websites include other tracking techniques running in
the background that can complement the information obtained for each
individual. Moreover, some of those systems are very intrusive tracking
methods, such as canvas or font fingerprinting. Thus, we can conclude that
currently all the websites complement the user information with other web
tracking sources to obtain a more precise profile of the user.

##  5 Discussion

Many actors have been exploring ways to enter the online advertising ecosystem
for years. The reason for this is simple: online advertisements are a major
source of income for many online companies. ISPs run the core network where
the Internet runs but, until now, they did not try to obtain economic profit
from the advertisement ecosystem running inside. Utiq is a step into this
direction, taking profit from the end of third-party cookies. It allows them
to capitalize on the growth in online advertising by leveraging their unique
position as the gateway to the Internet through the offer of precise user
identification, which is highly valuable to businesses. Nevertheless, they
offer the service, emphasizing its focus on privacy and transparency by
explicitly asking for the user’s consent in all cases and not saving personal
information about the browsing history. In practice, privacy-wise, the
combination of their identification method with other web tracking systems can
not only be as controversial as the third-party cookies, but even more because
of the possibility of precisely identifying the user over time.

On paper, Utiq seems like a good approximation to substitute the third-party
cookies used by many websites to identify the user. However, the adoption of
the system after six months is still very marginal, containing only a few
dozen websites. Thus, there must be a reason that holds other websites back
from integrating Utiq identification into their services. Although a
speculation, the fact that the user must accept two independent consent
managers, one from the main website and the other from Utiq’s service itself,
may interfere with the adoption of the system, as many websites were also
reluctant to introduce the cookie consent manager forced by the GDPR back in
2018 to not lose users. On the other hand, the exclusive agreement with AdForm
[24] may also be impacting their introduction, as there are many other
advertisers in the market that cannot compete for their bidding space because
of not being included.

##  6 Conclusions and Future Work

In this work, we took a first look at Utiq, a new identification method
proposed by some of the major internet service providers in Europe. The
identification is done based on two anonymized tokens extracted from a unique
identifier of the user, the internet connection. Our results show still a
marginal adoption of less than 1% after six months of operation. Although they
highly emphasize the private and transparent nature of the system, using this
methodology, companies can unambiguously identify the user over time. By
itself, the system could be a privacy-friendly approach for publishers and
advertisers, but when used in combination with other web tracking methods, it
may allow companies to create even more precise user profiling. Our study
shows that currently all websites including Utiq as one of their
identification methods also use other complementary web tracking systems, with
more than 80% of them using intrusive mechanisms such as fingerprinting
algorithms. Our future work includes studying if other identification
propagation techniques, such as cookie syncing, may be used with this kind of
unambiguous identification.

## Acknowledgements

This work was supported by the CHISTERA grant CHIST-ERA-22-SPiDDS-02
corresponding to the GRAPHS4SEC project (reference nº PCI2023-145974-2) funded
by the Agencia Estatal de Investigación through the PCI 2023 call.

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##  Appendix A Utiq compliant websites

Tranco rank |  Domain |  Country
---|---|---
786 |  lefigaro.fr |  France
841 |  bild.de |  Germany
936 |  elmundo.es |  Spain
1018 |  marca.com |  Spain
1065 |  ouest-france.fr |  France
1259 |  merkur.de |  Germany
1339 |  actu.fr |  France
1506 |  marmiton.org |  France
2385 |  welt.de |  Germany
2888 |  faz.net |  Germany
2956 |  focus.de |  Germany
3328 |  abc.es |  Spain
4454 |  eldiario.es |  Spain
4772 |  elperiodico.com |  Spain
5316 |  sudouest.fr |  France
5547 |  fr.de |  Germany
5693 |  sport.es |  Spain
6396 |  elcorreo.com |  Spain
6976 |  tz.de |  Germany
7059 |  hna.de |  Germany
7836 |  voici.fr |  France
7850 |  lachainemeteo.com |  France
9058 |  programme.tv |  France
10490 |  expansion.com |  Spain
11062 |  wiwo.de |  Germany
12455 |  utopia.de |  Germany
13732 |  farodevigo.es |  Spain
14685 |  mopo.de |  Germany
14992 |  lasprovincias.es |  Spain
15197 |  lne.es |  Spain
15302 |  diariovasco.com |  Spain
15409 |  levante-emv.com |  Spain
16089 |  elcomercio.es |  Spain
25772 |  aufeminin.com |  France
29822 |  ideal.es |  Spain
30343 |  laverdad.es |  Spain
30384 |  diariosur.es |  Spain
30903 |  eldiariomontanes.es |  Spain
33551 |  wa.de |  Germany
34097 |  caminteresse.fr |  France
34169 |  fnp.de |  Germany
35965 |  epe.es |  Spain
36443 |  diariodemallorca.es |  Spain
36637 |  kreiszeitung.de |  Germany
36949 |  hoy.es |  Spain
37321 |  elnortedecastilla.es |  Spain
38758 |  informacion.es |  Spain
39119 |  autoplus.fr |  France
39152 |  elperiodicodearagon.com |  Spain
40657 |  promiflash.de |  Germany
41891 |  laopiniondemurcia.es |  Spain
42416 |  eldia.es |  Spain
42919 |  laprovincia.es |  Spain
43756 |  laopiniondemalaga.es |  Spain
45457 |  larioja.com |  Spain
46415 |  diariocordoba.com |  Spain
47312 |  telva.com |  Spain
47428 |  elcorreogallego.es |  Spain
47496 |  elperiodicoextremadura.com |  Spain
47714 |  ingame.de |  Germany
47807 |  op-online.de |  Germany
49123 |  charentelibre.fr |  France
49246 |  laopinioncoruna.es |  Spain
50017 |  diariodeibiza.es |  Spain
50195 |  elperiodicomediterraneo.com |  Spain
51510 |  laopiniondezamora.es |  Spain
53081 |  superdeporte.es |  Spain
54247 |  maison-travaux.fr |  France
55751 |  diaridegirona.cat |  Spain
55823 |  larepubliquedespyrenees.fr |  France
56452 |  leonoticias.com |  Spain
60603 |  come-on.de |  Germany
64445 |  buzzfeed.de |  Germany
65374 |  regio7.cat |  Spain
65584 |  soester-anzeiger.de |  Germany
71058 |  eatbetter.de |  Germany
71971 |  bw24.de |  Germany
73728 |  landtiere.de |  Germany
73831 |  24vita.de |  Germany
82417 |  einfach-tasty.de |  Germany
90871 |  lavozdigital.es |  Spain
95878 |  atf-tagmanager.de |  Germany
97222 |  24auto.de |  Germany