ICTS

Branching random walk with regularly varying displacements is a heavy-tailed random field indexed by a Galton-Watson tree. We shall connect it to stable point processes introduced and characterized by Davydov, Molchanov and Zuyev (2008), who showed that such point processes can always be represented as a scale mixture of iid copies of one point process with the scaling points coming from an independent Poisson random measure. We obtain such a point process as a weak limit of a sequence of point processes induced by a branching random walk with regularly varying displacements. In particular, we show that a prediction of Brunet and Derrida (2011), remains valid in this setup, and recover a slightly improved version of a result of Durrett (1983). This talk is based on joint works with Ayan Bhattacharya and Parthanil Roy.

Computational statistical physics consists in, starting from a microscopic model of matter, evaluating macroscopic properties as ensemble averages. There are two types of quantities: (i) thermodynamic quantities are averages with respect to probability measures over the configurational space (canonical measure, microcanonial measure, etc...) ; (ii) dynamical quantities are obtained by sampling trajectories. Both are evaluated using probabilistic methods, as averages over stochastic processes. In terms of numerical computations, the difficulty to estimate these quantities is that the underlying stochastic processes are metastable: they remain trapped for very long times in some regions of the phase space, called metastable regions. Transitions between metastable regions are rare events. We will explain how this problem is circumvented in practice by appropriate numerical methods, and how to analyze mathematically these methods using various mathematical tools: entropy techniques, quasi-stationary distribution, semi-classical analysis, etc...

In several biological systems, phenotypic variations are seen even among genetically identical cells in homogeneous environments.  Recent research indicates that such `non-genetic individuality' can be driven by rare events arising from the intrinsic stochasticity of gene expression. Characterizing the fluctuations that give rise to such rare events motivates the analysis of large deviations in stochastic models of gene expression.

In this talk, I will discuss analytical approaches1 developed by my group for stochastic models of gene expression.  By developing a mapping to systems analyzed in queueing theory2, we derive analytical results characterizing mRNA and protein distributions for general kinetic schemes of gene expression. We combine approaches from queueing theory and non-equilibrium statistical mechanics to characterize large deviations and driven processes for general models of gene expression3. Modeling gene expression as a Batch Markovian Arrival Process (BMAP), we derive exact analytical results quantifying large deviations of time-integrated random variables such as promoter activity fluctuations. The results obtained can be used to quantify the likelihood of large deviations, to characterize system fluctuations conditional on rare events and to identify combinations of model parameters that can give rise to dynamical phase transitions in system dynamics.

References:

1. H. Pendar, T. Platini, and R. V. Kulkarni. “Exact protein distributions for stochastic models of gene expression using partitioning of Poisson processes” Phys Rev E, 87, 042720 (2013)
2. T. Jia and R.V. Kulkarni “Intrinsic noise in stochastic models of gene expression with molecular memory and bursting” Phys. Rev. Lett., 106, 058102 (2011)
3. J. Horowitz and R. V. Kulkarni “Stochastic gene expression conditioned on large deviations” Phys. Biol. Lett. (accepted for publication) (2017)

The McKean-Vlasov dynamics is a large system limit of the evolution of empirical measures of a weakly interacting system of particles. The empirical measure creates a field which then influences the evolution of any single particle. The process of empirical measures is itself a Markov process which has relative entropy with respect to the invariant measure as a natural Lyapunov function. By scaling this Lyapunov function and taking the infinite particle limit, we propose to arrive at a local Lyapunov function for the McKean-Vlasov dynamics. The approach is conceptually simpler than a prior approach, naturally identifies the desired Lyapunov function, and clarifies why the Freidlin-Wentzell quasipotential associated with the interacting system is a local Lyapunov function. (The talk is based on ongoing work with Laurent Miclo.)

We present a general method to derive a reduced description of a Markov chain which exhibits a metastable behavior.

Resetting in the context of stochastic processes is a way to model catastrophes in birth-death processes or the clearing of queues due to failure of a server in queueing theory. Brownian motion which is randomly reset to its starting position is a continuous space analogue and can be used to model search strategies where diffusive searching is combined with randomly returning to the starting point. (e.g. lost keys) In this talk I will introduce the mathematical formulation of resetting and show how this modifies the Fokker-Planck equation. The main result of the paper is a simple formula relating the Laplace transform in time of the generating function of time-additive observables of the process with resetting to the same object for the process without resetting. This formula makes it possible to study the large deviations of the process with resetting using the large deviations of the process without resetting which I will demonstrate for the Ornstein-Uhlenbeck process.

This is a joint work with Li-Cheng Tsai (Columbia University).
We prove a large deviation principle from the hydrodynamic limit of the TASEP, for deviations that have log-probability of order $-N^2$. The rate function is null on sub-solutions of the Burgers equation.

We investigated the efficiency of an isothermal Brownian work-to-work converter engine, composed of a Brownian particle coupled to a heat bath at a constant temperature. The system is maintained out of equilibrium by using two external time-dependent stochastic Gaussian forces, where one is called load force and the other is called drive force. Work done by these two forces are stochastic quantities. The efficiency of this small engine is defined as the ratio of stochastic work done against load force to stochastic work done by the drive force. The probability density function as well as large deviation function of the stochastic efficiency are studied analytically and verified by numerical simulations.

In thermodynamics, there exists a conventional belief that ``the Carnot efficiency is reachable only in the reversible (zero entropy production) limit of nearly reversible processes.'' However, there is no theorem proving that the Carnot efficiency is unattainable in an irreversible process. Here, we show that the Carnot efficiency is reachable in an irreversible process through investigation of the Feynman-Smoluchowski ratchet (FSR). We also show that it is possible to enhance the efficiency by increasing the irreversibility.Our result opens a new possibility of designing an efficient heat engine in a highly irreversible process and also answers the long-standing question of whether the FSR can operate with the Carnot efficiency.

Large deviations functions appear almost everywhere in Statistical Physics, in particular when one tries to quantify the frequency of rare events. Already the notion of free energy can be viewed as a large deviation function. Several well established properties of non equilibrium physics, such as the fluctuation theorem, can be formulated as a symmetry of a large deviation function. After a short review of these general properties, the talk will present a series of recent results on the large deviation function of the density and of the current of simple models of non equilibrium physics.

​These lectures will try to explain several approaches, both microscopic and macroscopic, to calculate the fluctuations and the large deviation functions of the current in non equilibrium diffusive systems. In steady state situations, one can obtain explicit expressions of these large deviation functions for diffusive systems. Less is known in non steady state situations or for non diffusive systems and the lectures will try to list a series of open questions.

Consider the problem of finding a population amongst many with the smallest mean when these means are unknown but population samples can be generated. Typically, by selecting a population with the smallest sample mean, it can be shown that the false selection probability decays at an exponential rate. Lately researchers have sought algorithms that guarantee that this probability is restricted to a small d in order log(1/d) computational time by estimating the associated large deviations rate function via simulation. We show that such guarantees are misleading. Enroute, we identify the large deviations principle followed by the empirically estimated large deviations rate function that may also be of independent interest. Further, we show a negative result that when populations have unbounded support, any policy that asymptotically identifies the correct population with probability at least 1-d for each problem instance requires more than O(log(1/d)) samples in making such a determination in any problem instance. This suggests that some restrictions are essential on populations to devise O(log(1/d)) algorithms with 1-d correctness guarantees. We note that under restriction on population moments, such methods are easily designed. Further, under similar restrictions, sequential methods from multi-armed bandit literature can also be adapted to devise such algorithms.

​These lectures will try to explain several approaches, both microscopic and macroscopic, to calculate the fluctuations and the large deviation functions of the current in non equilibrium diffusive systems. In steady state situations, one can obtain explicit expressions of these large deviation functions for diffusive systems. Less is known in non steady state situations or for non diffusive systems and the lectures will try to list a series of open questions.

The cover time is the time needed to visit all points on the lattice of size n. As a maximum of correlated random variables (here, with logarithmic decay) it has interesting asymptotics. In dimension 2, recurrence of the walk induces strong correlations, and the large deviations behavior is affected.

I consider the long time behavior of a random walker evolving in a one-dimensional, diffusive, random environment. If the environment is unbiased, the waker has no drift and the main question is about fluctuations. This problem turns out to be very much puzzling: the correlations of the environment decay slowly in time but their effect on the behavior of the walker is hard to predict. I will present a heuristic theory, supported by numerical results, that leads to predictions for the fluctuations of the walker as well as its differential mobility (response to an infinitesimal external force).

The Wiener sausage is the 1-environment of Brownian motion. It is an important mathematical object because it is one of the simplest non-Markovian functionals of Brownian motion. The Wiener sausage has been studied intensively since the 1970's. It plays a key role in the study of various stochastic phenomena, including heat conduction, trapping in random media, spectral properties of random Schr"odinger operators, and Bose-Einstein condensation.

In these lectures we look at two specific quantities: the volume and the capacity. After an introduction to the Wiener sausage, we show that both the volume and the capacity satisfy a downward large deviation principle. We identify the rate and the rate function, and analyse the properties of the rate function. We also explain how the large deviation principles are proved with the help of the skeleton approach.

Joint work with Michiel van den Berg (Bristol) and Erwin Bolthausen (Zurich).

In this talk we shall  introduce non-local operators that come from jump diffusions and prove a Harnack Inequality for the Schrodinger operator.  We shall present some open problems related to large deviations. This is joint work with Koushik Ramachandran

The Wiener sausage is the 1-environment of Brownian motion. It is an important mathematical object because it is one of the simplest non-Markovian functionals of Brownian motion. The Wiener sausage has been studied intensively since the 1970's. It plays a key role in the study of various stochastic phenomena, including heat conduction, trapping in random media, spectral properties of random Schr"odinger operators, and Bose-Einstein condensation.

In these lectures we look at two specific quantities: the volume and the capacity. After an introduction to the Wiener sausage, we show that both the volume and the capacity satisfy a downward large deviation principle. We identify the rate and the rate function, and analyse the properties of the rate function. We also explain how the large deviation principles are proved with the help of the skeleton approach.

Joint work with Michiel van den Berg (Bristol) and Erwin Bolthausen (Zurich).

Systems driven out of equilibrium often reach stationary state which under generic conditions exhibit long-range correlation. Such correlations have been observed both in experiment in fluid medium as well as in theoretical models. I shall present a fluctuating hydrodynamics description of interacting particles in which both spatial and temporal correlation can be obtained systematically to arbitrary order. The formulation is an application of the Macroscopic Fluctuation Theory. The long-range nature of the correlation can be seen in an electrostatic analogy where the correlation is potential due to localized charge which is non-zero when detailed balance is violated. To verify the hydrodynamic results, I shall derive spatial and temporal correlations in an exact solution of the symmetric exclusion process. Our exact solution also presents a direct verification the fluctuating hydrodynamics equation. As an application of our results, I shall discuss a generalization of the fluctuation dissipation relation in non-equilibrium stationary state where the response function is expressed in terms of two-time correlations.

References:

1. Tridib Sadhu and Bernard Derrida, Correlations of the density and of the current in non-equilibrium diffusive systems, J Stat Mech (2016) 113202.
2. Tridib Sadhu, Satya N Majumdar, and David Mukamel, Long-range correlations in a locally driven exclusion process, Phys. Rev. E. 90, (2014) 012109.

I will give in these lectures an overview of techniques used in large deviation theory to study the fluctuations of time-additive quantities, called dynamical observables, for Markov processes (jump processes or diffusions) modelling equilibrium and nonequilibrium processes driven by external forces and noise. These fluctuations are described by large deviation functions, obtained by solving a dominant eigenvalue problem similar to the problem of finding the ground state energy of quantum systems. This will be covered in the first lecture. In the second lecture, I will discuss more recent research describing how fluctuations of dynamical observables are created in time by means of an 'effective' process, called the driven process. This process can be seen as a process generalisation of the concept of fluctuation paths, introduced by Onsager and Machlup (and later Freidlin and Wentzell) to describe noise-activated transitions in the low-noise limit.

We analyze large deviations of the time-averaged activity in the one-dimensional Fredrickson-Andersen model, both numerically and analytically. The model exhibits a dynamical phase transition, which appears as a singularity in the large deviation function. We analyze the finite-size scaling of this phase transition numerically, by generalizing an existing cloning algorithm to include a multicanonical feedback control: this significantly improves the computational efficiency. Motivated by these numerical results, we formulate an effective theory for the model in the vicinity of the phase transition, which accounts quantitatively for the observed behavior. We discuss potential applications of the numerical method and the effective theory in a range of more general contexts. Work in collaboration with Takahiro Nemoto, Robert L. Jack

I will give in these lectures an overview of techniques used in large deviation theory to study the fluctuations of time-additive quantities, called dynamical observables, for Markov processes (jump processes or diffusions) modelling equilibrium and nonequilibrium processes driven by external forces and noise. These fluctuations are described by large deviation functions, obtained by solving a dominant eigenvalue problem similar to the problem of finding the ground state energy of quantum systems. This will be covered in the first lecture. In the second lecture, I will discuss more recent research describing how fluctuations of dynamical observables are created in time by means of an 'effective' process, called the driven process. This process can be seen as a process generalisation of the concept of fluctuation paths, introduced by Onsager and Machlup (and later Freidlin and Wentzell) to describe noise-activated transitions in the low-noise limit.

The effect of quenched disorder on the behavior of a system, whose pure version undergoes a first order transition is well understood by now for d<=2. For two dimensions, an infinitesimal amount of disorder is known to change the order of transition to a continuous transition. The situation in higher dimensions is not so clear. In higher dimensions, it was conjectured that there exist a impurity threshold beyond which the transition should change its character[2]. We solve $p$ spin interaction model and Blume Capel model in the presence of random field disorder on fully connected graphs using large deviation theory. For $p$ spin interaction model, we find that for p>= 3, the first order transition continues to be a first order for all strengths of disorder. While for Blume Capel model with bimodal random crystal field, we find the behavior is modified: for very weak disorder, the phase diagram is similar to the pure model with regions of first and second order transitions separated by a tricritical point. With increasing strength of disorder, the transition soon becomes a continuous transition.

Feynman-Kac semi-groups naturally appear in the context of large deviations. In particular, their long time behavior provides the scaled cumulant generating function, which is the dual of the rate function. However, their practical implementation is difficult in practice. Several work address their statistical approximation but, for diffusions, one also has to discretize the process in time. In this context, I will present a framework based on Talay-Tubaro expansion that provides error estimates on the long-time behavior of such discretized processes. We will see that a novelty of this work is to take into account the non-probability conserving feature of the dynamics, which provides an alternative expression for the cumulant function. Our analysis is supported by numerical applications.

This talk will be divided in two parts. In the first, I will discuss a general theory that allows to compute perturbatively in a small parameter the non-equilibrium free energy of many-body systems, for example when the system is not too far from equilibrium. I will then apply it to two cases: mean-field interacting diffusions and a minimal model of self-propelled particles. In this second case, our approach allows to show that the system is described by a non Botzmann-Gibbs steady state but, at the same time, entropy production vanishes. In the second part of the talk, I will concentrate on field theories which describe phase-separation in systems composed of self-propelled particles, such as bacteria or active colloids. These field theories are minimal modification of Model B (stochastic Cahn-Hilliard equation). I will discuss that even a small amount of activity, i.e. a minimal perturbation from an equilibrium dynamics, can strongly modify the nucleation dynamics.

We consider dynamical large deviations in Markov chains.  The associated rate functionals are not quadratic, leading to non-linear flux-force relations. We discuss a canonical decomposition of the rate function based on conjugate forces and fluxes, and we also split the `force' into a time-reversal-symmetric (equilibrium) part, and an anti-symmetric (non-equilibrium) part [1].  These two parts of the force satisfy a generalised orthogonality condition. We discuss the implications of these results, and their connections to Macroscopic Fluctuation Theory [2] and to previous work by Maes and co-workers [3].

References:
[1] Kaiser, Marcus, Robert L. Jack, and Johannes Zimmer, arXiv:1708.01453 (2017).
[2] Bertini, Lorenzo, et al., Reviews of Modern Physics 87.2 (2015): 593.
[3] Maes, Christian, and Karel Netočný, EPL (Europhysics Letters) 82.3 (2008): 30003.

Conventional phase transitions involve abrupt changes of a macroscopic system in response to small variations of an external control parameter. This exceptional behavior can be understood from the complex zeros of the partition function of the finite-sized system: in the thermodynamic limit, these Lee-Yang zeros, which correspond to logarithmic singularities of the free energy, approach the critical value of the control parameter on the real axis. This general scheme also applies to dynamical phase transitions in non-equilibrium systems. The partition function is thereby replaced with the moment-generating function of a stochastic process with the counting field playing the role of the external control parameter. Here, we demonstrate that the corresponding dynamical Lee-Yang zeros are not only a theoretical concept but physical observables, which encode remarkable information on the long-time statistics and the dynamical fluctuations of the system [1]. To this end, we analyze a stochastic process involving Andreev-tunneling events in a mesososcopic device consisting of a normal-state island and two superconducting leads. From measurements of the dynamical activity, we extract the Lee-Yang zeros, which reveal a smeared dynamical phase transition outside the range of direct observations. Being obtained only from short-time data, this information allows us to predict the large-deviation statistics of the dynamical activity at long times, which is otherwise difficult to measure. Our method paves the way for further experiments on the statistical mechanics of many-body systems out of equilibrium.

This series of lectures will focus on large deviations for random variables with power law (or more generally, regularly varying) distributions. Because of polynomial decay of the tail, rare events happen in many cases due to the large value of one random variable. This phenomenon is called one large jump principle, which will be explained in these lectures. Major difference with classical large deviation theory for light-tailed distributions will also be discussed.

Markov jump processes can generate steady-state probability currents at the expense of dissipation. I will discuss how the dissipation also constraints the fluctuations in those currents. A small deviations corollary proves a "thermodynamic uncertainty principle"---to reduce the uncertainty in the estimate of nonequilibrium currents, a process must dissipate more. I will further discuss corresponding dissipation-based uncertainty bounds for first passage time fluctuations.

This series of lectures will focus on large deviations for random variables with power law (or more generally, regularly varying) distributions. Because of polynomial decay of the tail, rare events happen in many cases due to the large value of one random variable. This phenomenon is called one large jump principle, which will be explained in these lectures. Major difference with classical large deviation theory for light-tailed distributions will also be discussed.

Large deviation nature is characterized by non-Gaussian statistics. The simplest model to characterize such systems is the motion of particles activated by non-Gaussian white noise. I also stress the effects of roles of non-analytic friction such as Coulombic dry friction. In the former half part, I am going to talk about the relevant condition for the non-Gaussian noise which means the violation of the central limit theorem even in long time limit observation of noises. In the later half part, I will talk on many-body effects of particles activated by non-Gaussian noise, in which effective attractive interaction appears.

I will review recent developments in applying dynamical large deviations (LD) to quantum systems. I will consider in particular open quantum systems - quantum systems interacting with an environment - which in many cases can be described in terms of quantum Markovian dynamics. LD methods provide a "thermodynamic" framework for understanding the statistical properties of dynamics, revealing the existence of dynamical phases and phase transitions, often associated to intermittent emission patterns. Problems of interest include interacting atomic ensembles, quantum glasses, and systems where there is an interplay between coherent transport and dissipation. I will describe concepts relating to quantum trajectory ensemble equivalence, prediction and "retrodiction", matrix product states, and quantum Doob transforms. Time permitting, I will also discuss the connection to ideas about slow quantum relaxation and metastability.

We compare dynamical fluctuations in stochastic processes with and without detailed balance. For general Markov chains, there are explicit rate functions for the joint large deviations of the empirical current and density, which can be derived either in a long-time limit (so-called level-2.5) or by considering many copies of the Markov chain (an ensemble) [1]. For some Markov processes, such as exclusion processes, the behaviour on large length and time scales can be described by macroscopic fluctuation theory (MFT) [2]. This theory also provides formulae for rate functions of the current and density, as well as a decomposition of the current into two orthogonal contributions that are symmetric and anti-symmetric under time reversal. In the context of sampling by Markov chain Monte Carlo (MCMC), a number of recent results indicate that systems where detailed balance is broken generically converge more rapidly to their steady states, leading to improved sampling efficiency [3]. We show how this result is related to the existence of two orthogonal currents within MFT [4]. Then, we show how a similar pair of currents can be defined in general Markov chains, together with their conjugate forces [5]. This last result shows how some aspects of MFT are already present within generic Markov chains, as well as providing new insight into the improved sampling efficiency that is available on breaking detailed balance.

[1] Maes, Netocny, Wynants, Markov Proc. Rel. Fields 14, 445 (2008)
[2] Bertini, De Sole, Gabrielli, Jona-Lasinio, Landim, Rev Mod Phys 87, 593 (2016)
[3] Rey-Bellet, Spiliopoulos J Stat Phys 164, 472 (2016)
[4] Kaiser, Jack, Zimmer, J Stat Phys 168, 259 (2017)
[5] Kaiser, Jack, ZImmer, in preparation.

Small systems in contact with a heat bath evolve by stochastic dynamics. We show that, when one such small system is weakly coupled to another one, it is possible to infer the presence of such weak coupling by observing the violation of the steady state fluctuation theorem for the partial entropy production of the observed system. We give a general mechanism due to which the violation of the fluctuation theorem can be significant, even for weak coupling. We analytically demonstrate on a realistic model system that this mechanism can be realized by applying an external random force to the system. In other words, we find a new fluctuation theorem for the entropy production of a partial system, in the limit of weak coupling.

TBA

I shall discuss a simple exact description of non-equilibrium states with non-uniform temperature (or chemical potential) profiles in (1+1)D conformal field theory.

Many natural systems are far from thermodynamic equilibrium and keep on exchanging matter, energy or information with their surroundings. These exchanges produce currents, or fluxes, that break time-reversal invariance. Such systems lie beyond the realm of traditional thermodynamics and the principles of equilibrium statistical mechanics do not apply to them. In fact, there exists no general conceptual framework a la Gibbs-Boltzmann to describe these systems from first principles.

The last two decades, however, have witnessed remarkable progress. The aim of this lecture is to explain some recent developments, such as the Macroscopic Fluctuation Theory (MFT) of Bertini, Gabrielli, Da Sole, Jona-Lasinio and Landim. This theory represent the first steps towards a unifi ed approach to non-equilibrium behaviour.

After reviewing the approach of Onsager and Machlup of fluctuations in the vicinity of equilibrium, we shall explain the basis of the MFT from a "user's approach". We shall give examples of lattice gases for which the transport coefficients can be exactly evaluated and describe some applications to the calculations of large deviation functions of currents and density profiles.

Many natural systems are far from thermodynamic equilibrium and keep on exchanging matter, energy or information with their surroundings. These exchanges produce currents, or fluxes, that break time-reversal invariance. Such systems lie beyond the realm of traditional thermodynamics and the principles of equilibrium statistical mechanics do not apply to them. In fact, there exists no general conceptual framework a la Gibbs-Boltzmann to describe these systems from first principles.

The last two decades, however, have witnessed remarkable progress. The aim of this lecture is to explain some recent developments, such as the Macroscopic Fluctuation Theory (MFT) of Bertini, Gabrielli, Da Sole, Jona-Lasinio and Landim. This theory represent the first steps towards a unifi ed approach to non-equilibrium behaviour.

After reviewing the approach of Onsager and Machlup of fluctuations in the vicinity of equilibrium, we shall explain the basis of the MFT from a "user's approach". We shall give examples of lattice gases for which the transport coefficients can be exactly evaluated and describe some applications to the calculations of large deviation functions of currents and density profiles.

In this talk, I will speak on second principle and Landauer bound. Despite some old and classical formulations, some points are still source of interest and debates. By example the physical contents of the Gibbs-Shannon entropy outside equilibrium. In peculiar, I will explain the theoritical background under recent experimental measure of the Gibbs-Shannon Entropy associated to a bit by Gavrilov-Chetrite-Bechhoefer.

The large deviation theory describing rare events forms a mathematical basis for statistical physics. In equilibrium situations, the rate function estimating ``rareness” gives the entropy function. On the other hand, in nonequilibrium situations, we can evaluate the entropy production by calculating the rate function on the time series describing the time evolution of nonequilibrium dynamics. Beyond the field of physics, it has been recently suggested that the large deviation theory can be applied to analysis of growing cell populations. To be more precise, the stationary population growth rate (also called ``fitness”) can be evaluated by the Legendre transform of the rate function on the cell lineage. Owing to this structure, we can calculate responses of the population growth with respect to external environmental changes. Furthermore, it is elucidated that such responses are also evaluated by statistics on a lineage obtained by retrospectively tracing ancestors. In this talk, we demonstrate the above framework on an age-structured population model, which is asexual population dynamics with age- and type-dependent growth rate. In a calculation for this population model, the explicit form of the rate function for the semi-Markov process plays an essential role. In the first half of this talk, we briefly show the explicit form of the rate function that is an extension of the ``Level 2.5” rate function on Markov processes to semi-Markov situations. In the second half, by using the explicit form, we demonstrate the path-wise analysis for the age-structured population model.