Dr Andrea Rocco

In this research theme, I explore the origin of the arrow of time and irreversibility in thermodynamic systems. While our everyday experience tells us that time flows from the past towards the future, the fundamental equations describing the universe are, to a large extent, time symmetric: if we reverse the direction of time, backward-in-time trajectories are just as admissible as forward-in-time ones. How, then, can an arrow of time emerge from underlying time-symmetric dynamics?

Open quantum systems (OQSs), quantum systems interacting with their environment, offer a natural framework in which to address this question. Their interaction with the environment allows energy and quantum information to be dissipated away from the system of interest. One might therefore expect this process to generate positive entropy production and, correspondingly, to break time-reversal symmetry in the reduced dynamics of the system.

In my group, however, we have shown that the situation is more subtle. A careful derivation of the reduced OQS dynamics indeed reproduces irreversible behaviour in the Markovian limit, as manifested by classical and quantum entropies that increase monotonically with time. Surprisingly, however, the reduced equations of motion themselves can retain time-reversal invariance. This symmetry acts about the time t = 0 at which the Markovian description is introduced. As a consequence, the system can thermalize in two opposite directions of time emerging symmetrically from t = 0, displaying irreversible behaviour both towards the future and towards the past [Guff et al., Scientific Reports, 2025].

We are now investigating the implications of these findings and, in particular, seeking to identify additional dynamical mechanisms capable of selecting a unique direction of time from the two directions that are, in principle, equally admissible.

Quantum systems do not exist in isolation. When a quantum system interacts with its environment, it becomes an open quantum system (OQS) and typically undergoes decoherence. Decoherence is one of the most fascinating concepts in fundamental physics: it lies at the heart of understanding how the strange world of quantum mechanics gives rise to the familiar macroscopic world of our everyday experience. Yet precisely how classical behaviour emerges from the underlying microscopic quantum dynamics remains a profound question.

Describing the dynamics of OQSs is challenging. It typically involves projection techniques and controlled approximations aimed at deriving a quantum master equation for the reduced density matrix of the system. The resulting reduced dynamics provide a framework for investigating dissipation and decoherence, the quantum-to-classical transition, and the emergence of irreversible behaviour at both the quantum and classical levels.

In my group, we study non-Markovian dynamics, which arise when there is insufficient separation between the characteristic timescales of the system and its environment, so that memory effects become important in the reduced dynamics. Our results show that these effects can substantially modify the decoherence process, including through the emergence of what we call "lateral coherences" [Lally et al., Phys. Rev. A, 2022]. We also investigate how different environmental structures affect the resulting reduced dynamics. Recent work in my group has revealed the emergence of anomalous thermodynamic properties in models with hierarchical environments [Guff & Rocco, Phys. Rev. E, 2026].

We also investigate the quantum-to-classical (QC) transition as a critical phenomenon. To do so, we adopt a Renormalization Group (RG) approach, extending it to open quantum systems in order to identify the scaling properties of their reduced dynamics. Our recent findings indicate that, alongside quantum dissipation, the low-frequency behaviour of decoherence must also be taken into account to fully characterize the QC transition. The resulting RG flow, governed by a generalized Wegner-Houghton equation, describes the progressive elimination of quantum fluctuations, with the classical limit expected to emerge as a fixed point of the flow.

Molecular biological systems, such as gene regulatory and metabolic networks, are inherently subject to stochastic fluctuations. Fluctuations arising from low copy numbers of molecules or from the stochastic nature of gene expression are generally referred to as intrinsic noise, while fluctuations originating from the cellular environment are commonly described as extrinsic noise. In this strand of my research, I investigate both forms of noise, and their interplay, using Langevin and Fokker-Planck approaches.

Extrinsic noise can produce highly non-trivial and sometimes counterintuitive effects, including noise-induced transitions [Rocco et al., Springer (2013)], a phenomenon encountered across a variety of physical and chemical systems. In the case of multiplicative Gaussian white noise, such effects are closely related to the Stratonovich drift, through which fluctuations can modify the effective deterministic dynamics and alter the number and stability of its stationary states. Building on these ideas, I introduced the concept of stochastic control in metabolic networks [Rocco, Phys. Biol. (2009)], whereby noise itself can act as a control mechanism, tuning metabolic concentrations and fluxes.

In my group, we have subsequently developed a dynamical framework for describing bounded, non-Gaussian and nonlinear extrinsic fluctuations in gene expression [Aquino & Rocco, Math. Biosci. Eng. (2020)]. Such fluctuations can also generate noise-induced transitions, but through mechanisms that differ substantially from those associated with Gaussian white noise. We have shown, for example, that these effects can help account for the heterogeneity in growth and drug tolerance observed within clonal bacterial populations [Rocco et al., PLOS ONE (2013); Hingley-Wilson et al., PNAS (2020)].

More recently, we developed a formalism that relaxes the assumption of an infinitely slow environment, allowing us to describe extrinsic fluctuations whose characteristic timescales are slow, but not infinitely separated from those governing transcriptional and translational dynamics [Aquino & Rocco, Phys. Rev. E (2023)]. This provides a more general framework for investigating how environmental fluctuations propagate through gene-expression dynamics and influence cellular behaviour.

A major problem in developmental biology is to understand how distinct cell types emerge from multipotent stem cells. This problem is particularly well suited to Dynamical Systems Theory, which provides a natural mathematical framework for describing cellular decision-making and fate selection in terms of attractors, stability and bifurcations.

In collaboration with experimentalists at the University of Bath, we constructed the first core gene regulatory network describing stable melanocyte differentiation in zebrafish [Greenhill et al., PLoS Genetics (2011)]. Combining mathematical analysis with numerical simulations, we predicted several previously unknown features of the network that were subsequently validated experimentally. We later refined this core network to investigate the role of Wnt signalling in melanocyte differentiation [Vibert et al., Pigment Cell & Melanoma Research (2017)] and extended the framework to other neural-crest-derived pigment cell types [Petratou et al., PLoS Genetics (2018)].

More recently, our research has focused on the possible role of cyclic dynamics in cell differentiation. We introduced a new framework [Kelsh et al., Development (2021)] in which a multipotent cell does not simply occupy a single stable state, but instead cycles through a sequence of metastable states. While transiently occupying any one of these states, the cell may respond to external signals that interrupt the cycle and drive it towards a differentiated fate [Farjami et al., J. Royal Soc. Interface (2021)]. Detailed experimental analysis of direct versus progressive fate-restriction models in zebrafish pigment cells provides evidence consistent with this hypothesis [Subkhankulova et al., Nature Communications (2023)].

We have recently analysed in detail the bifurcation structure underlying these cyclic dynamics [Schofield et al., Proc. R. Soc. (2026)] and are now investigating how stochastic fluctuations interact with the cycle to influence cell-fate decisions.