Krylov complexity of open quantum systems: from hard spheres to black holes

We examine the complexity of quasi-static chaotic open quantum systems. As a prototypical example, we analytically compute the Krylov complexity of a slowly leaking hard-sphere gas using Berry’s conjecture. We then connect it to the holographic complexity of a d + 1-dimensional evaporating black hole using the Complexity=Volume proposal. We model the black hole spacetime by stitching together a sequence of static Schwarzschild patches across incoming negative energy null shock waves. Under certain identification of parameters, we find the late time complexity growth rate during each quasi-static equilibrium to be the same in both systems.


Introduction
In recent years, the complexity of quantum systems has shown much promise as a tool in diagnosing quantum chaos [1][2][3][4][5].Complexity tracks the degrees of freedom of the system, especially at very long time scales where other quantum information theoretic measures like entanglement entropy have saturated [6].This feature has been particularly fruitful in theories involving black holes, where the volume of spacelike extremal codimension-1 surfaces has been conjectured to measure the complexity of the corresponding state in the dual boundary theory [7,8].The proposal, dubbed the Complexity=Volume (CV) prescription, adds a non-trivial entry to the holographic dictionary.
In a quantum mechanical system, we can quantify complexity using Krylov Complexity (or K-complexity) [1,9], which measures the growth of operators by treating its evolution as a particle hopping on a semi-infinite chain 1 .Under Hamiltonian evolution, a wavefunction centered around the first site will spread deeper into the chain.Krylov complexity is defined as the average position of the particle on the chain as a function of time, thereby capturing the "spread" of the operator in the operator space.
In section 2, we will use this formulation of operator complexity to study the growth of operators in a slowly leaking hard sphere gas.We will assume that the gas is leaking from a small box into a bigger box, as in [24].Moreover, we will work in the semiclassical, low density limit of the system.A box of hard sphere gas is classically chaotic and satisfies Berry's conjecture, which states that the high-lying energy eigenstates behave as if they have been picked from a Gaussian ensemble [25][26][27].Berry's conjecture played a crucial role in arriving at Eigenstate Thermalization Hypothesis (ETH) [27] (see also [28]) and makes the system analytically tractable.
However, since we are dealing with an open quantum system, the canonical Krylov complexity calculations would not work here.This is because the evolution is non-unitary, and the Lanczos algorithm one usually employs to calculate the Kcomplexity leads to unsatisfactory results [29][30][31][32].We will sidestep this difficulty by working with a slowly leaking gas.This allows us to focus on a time period, which we will refer to as an epoch, during which the boxes equilibrate separately, and there is no overall exchange of particles.During an epoch, the Lanczos algorithm gives meaningful results, and we can compute the Krylov complexity using standard techniques.Moreover, we will also show that we can patch together adjacent epochs to form a continuous curve.
Using ETH and very mild assumptions on the off-diagonal matrix elements of the operators, we will argue how the hard sphere calculation can be generalized to any chaotic open quantum system.This prompts us to look at our slowly leaking hard sphere gas model as an excellent prototype where analytic calculations can be carried out explicitly.
In section 3, we will use the CV prescription to calculate the holographic complexity of an evaporating black hole.We will model an evaporating black hole by stitching together a sequence of static Schwarzschild spacetimes across incoming negative energy null shock waves.Each Schwarzschild patch is characterized by a constant mass that decreases as we go across the shock waves.These patches correspond to periods where the black hole is effectively not evaporating.Therefore, we will refer to them as epochs, akin to our slowly leaking gas calculation.We will calculate the volume of boundary-anchored extremal codimension-1 surfaces in this background.Under an identification of parameters, we show that the late time rate of growth of complexity during an epoch matches the slowly leaking gas calculation.

Krylov Complexity of a Slowly Leaking Gas
Consider an operator O that acts on the states of a quantum mechanical system.If the Hamiltonian of the system is H, then the time evolution of this operator is given by where L(O) = [H, O] is the Liouvillian superoperator.Krylov complexity measures the spread of O(t) in the Krylov subspace, the Hilbert space spanned by L n O.We will not provide a pedagogical review of Krylov complexity here as detailed reviews can be found elsewhere [1,9,33].
The microcanonical refinement of Krylov complexity, as introduced in [34], will be our focus for analysis.Additionally, we will use the moment method to compute Krylov complexity.Let us quickly review this construction.Our starting point is the thermal two-point function of the operator: Here β is the inverse temperature, and E i,j are the energy eigenvalues of the system.Approximating the sum by an integral over the density of eigenstates ρ(E), we get where we have defined the average energy and the energy difference as follows: The Liouvillian is sensitive only to the energy differences ω, and it does not mix different average energy E sectors [34].Therefore, we can work with a fixed E and then average over all the other energy sectors at the end of the calculation.The fixed energy two point function can be obtained by taking its inverse Laplace transform: (2.5) The key elements in our analysis are the moments of these two point functions.They are given by Using the Hankel transformation matrix The Lanczos coefficients are handy objects because they contain all the information about the dynamics of the operator O.Moreover, they completely determine the Krylov complexity of the operator.To see this, let us note that the fixed energy Krylov complexity is given in terms of the K-wavefunctions ϕ E,n as follows [34] where D O is the dimensionality of the Krylov subspace.The K-wavefunctions are, in turn, related to the Lanczos coefficients through the Schrödinger equation: (2.9) Using the initial condition ϕ E,n (0) = δ n0 , we can solve for the K-wavefunctions and compute K E .Krylov Complexity K E measures the complexity growth of operators within a specific energy sector.Operationally, this definition of complexity corresponds to the choice of a "microcanonical" inner product on the Krylov subspace: (2.10) If we had instead used a thermal inner product that involved integrating over various energy sectors, there would have been a risk of the signatures of chaos being washed out (See [34] for details).Therefore, the natural thing to do is to compute the growth in a fixed energy sector and then take an average over various values of E. Since different energy sectors are thermally populated, we can define the thermal Krylov complexity by taking a Laplace transform of its fixed energy counterpart [34]: where the normalization constant C(E) is given by (2.12) Our primary focus is on the thermal Krylov complexity.In the following subsections, we will use the moments of the thermal two point functions to calculate K th (t) of a slowly leaking hard sphere gas.

Warm-up: Single Box
Before we look at the slowly leaking gas, it is instructive to look at a single box of hard sphere gas.Consider a cubic box of edge length L + 2a enclosing N hard spheres.We will assume that the hard spheres are identical and have radius a.The classical Hamiltonian of the system is given by where This system is classically chaotic and shows eigenstate thermalization when treated quantum mechanically [27].Let us denote the energy eigenfunctions of the system by ψ(X), where X is the 3N -dimensinal position vector.The wavefunctions are defined on the domain We will impose the boundary condition that ψ(X) vanishes on ∂D.The wavefunctions with energy E α can be chosen to be of the following form [27]: where N α is the normalization constant.We can choose the wavefunction to be everywhere real by imposing A * α (P) = A α (−P).Let us define the thermal wavelength λ of the system as follows: where the termperature T α is related to the energy through the relation E α = 3 2 N kT α .When the thermal wavelength λ ≲ a, then the high-lying energy eigenstates are expected to satisfy the Berry's conjecture, which states that A α (P) can be regarded as a Gaussian variable with the two-point function ⟨A α (P)A β (P ′ )⟩ EE = δ αβ δ 3N (P + P ′ ) δ (P 2 − P ′2 ) . (2.18) The subscript EE stands for eigenstate ensemble, the fictitious Gaussian ensemble the high-lying wavefunctions can be thought to belong to.The wavefunction in the momentum space is given by: We will assume that we always work in the regime where Berry's conjecture holds (λ ≲ a).Moreover, we will also assume that the density of the gas is low, that is, N a ≪ L 3 .These assumptions give us enormous analytic control over the system.In particular, it is easy to see that the averaged two point functions are given by [27] where (2.21)

Krylov Complexity from Moments
Consider the operator P 1 that measures the momentum of one of the particles.The matrix elements of this operator in the energy eigenbasis are given by where |p 1 | is the magnitude of the momenta of one of the particles.Let us calculate the Krylov complexity of this operator by using the moments of the thermal two point function.Using (2.6), we get C(E) is the normalization constant we defined in (2.12).The density of eigenstates ρ is given by [27] ρ (2.24) Now let us calculate the average of the moments in the eigenstate ensemble: Here we have used the shorthand notation ρ 0 (E, ω) for the product of the density of states.Using (2.22), we see that: The four-point function can be broken down into two point functions using Wick contractions: From (2.20), we can see that contracting two eigenfunctions will produce a delta function in its indices.The only terms in (2.27) which would contribute to the moment calculation are the ones without any δ mℓ factor.This is because (2.25) contains a factor (E ℓ − E m ), multpyling the four-point functions.Therefore, only the last term in (2.27) would contribute.Let us look at the integral of this term separately: Using (2.20), we see that where we have used [27] (δ 3N Now let us look at the m = ℓ component.This corresponds to the case where ω = 0 and E m = E ℓ = E.In the low density limit, we can essentially replace δ 3N D (P − P ′ ) with a dirac delta.This gives us Choosing the normalization constant as in [27] we get Now let us return to the non-diagonal elements of Φ mℓ .From our definition (2.21), it is easy to see that δ 3N D (P − P ′ ) is a sharply peaked function that is zero almost everywhere.Let us choose this function to be a Gaussian distribution as in [27]: This gives us [27] Φ In the notation Plugging in the expressions, we find that the following integral gives the moments:

A Tale of Two Saddles
We will compute the moment integral (2.37) using a saddle point approximation.We claim that for small moments n, the saddle point ω * is located at To see this, let us work out the saddle point equation up to the leading order in ω/2E.This gives us Solving the above equation, we find that the saddle point is located at where λ is the thermal wavelength at the energy E. When n ≪ 3N L 2 2λ 2 , ω * is much smaller than 2E, as advertised.Substituting the saddle point in (2.37), we get: When n is sufficiently large, we can calculate the Lanczos coefficients using the relation [1,9,34] We find that This behavior is termed "Lanczos ascent" in literature [34].When n ≫ 3N L 2 2λ 2 , ω * ≫ 2E.Since this violates our assumption (2.38), (2.40) ceases to be a saddle point of the moment integral.For large n, (2.40) gets replaced by a new saddle point located at ω * ≃ 2E.We can obtain this saddle point mechanically by noting that if the saddle point equation is given by Solving the equation, we find that the new saddle point is located at When It is easy to verify that (2.46) satisfies (2.44) when n ≫ 3N L2 2λ 2 .Therefore, the saddle point of the integral changes when we look at higher moments.This has interesting consequences for the behavior of Lanczos coefficients.In particular, we can use (2.42) to see that: (2.48) The saturation of Lanczos coefficients is referred to as "Lanczos plateau".This gives us From [9,35,36], we can see that these Lanczos coefficients result in an initial scrambling phase where the Krylov complexity grows quadratically.Following the scrambling phase, K-complexity switches to linear growth.Working out the growth rates and reinstating factors of ℏ, we find that the Krylov complexity is given by where t S is the scrambling time.We can determine t S by noting that the dynamics of the operator can be thought of as a particle moving on a one-dimensional semiinfinite chain.The sites on the chain are labeled by n, and Krylov complexity (2.8) measures the average position ⟨n⟩.The saddle points change when n ∼ 3N L 2 2λ 2 ≡ n S .Therefore, scrambling time is the amount of time Krylov complexity takes to reach n S .This gives us Using (2.11), we can compute the thermal K-complexity in the eigenstate ensemble: It is easy to see that we get where E * and λ * are the average energy and thermal wavelength of the box.Plotting these functions, we obtain figure 1.

Slowly Leaking Gas
Now let us look at the slowly leaking gas model used in [24].Consider two cubic boxes sharing a common side as in figure 2. Let us assume that the left box has N hard spheres while the right box is empty.At time t = 0, we make a small hole in their shared wall so that the gas leaks slowly into the right box.We assume that the gas is leaking so slowly that there exists a time scale over which both the boxes have separately equilibrated.We will refer to this period as an epoch.During each epoch, the number of particles in each box, denoted by N L,R , remains roughly constant.Since we are working in the semiclassical limit, we can always use either N L or N R to characterize each epoch.

Krylov Complexity during an epoch
Within an epoch, there is no net exchange of particles between the boxes.Consequently, when examining the full Hamiltonian, it becomes evident that during each epoch, the Hamiltonian takes on the following factorized structure: Consider an operator P 1 , which measures the momentum of one of the particles in the left box.The operator has the following form

.55)
Here P 1,L is an operator that acts only on the left box.Now let us look at the Krylov complexity of this operator.If we are computing the Krylov complexity w.r.t to the entire system, we will get the results of the previous section -a scrambling phase followed by linear growth.This is because the left and right boxes form a closed classically chaotic system, and we anticipate a version of Berry's conjecture to be applicable to the eigenstates of the entire system.However, we are after the Krylov complexity of the left box alone, which is an open quantum system.The evolution of an operator O L acting on the left box is given by the master equation: Here, L L denotes the Liouvillian of the left box, and L D represents a dissipative term arising from the interaction between the two boxes.The dissipative term makes the evolution non-Hermitian, and there is no consensus on extending Krylov complexity calculations to open quantum systems [29][30][31].However, the "effective" Hamiltonian has no interaction term during an epoch.Therefore, the evolution of an operator O L would be controlled only by L L , or equivalently, the Hamiltonian of the left box.This allows us to carry over our definition of moments (2.23) to the two-box system: where E L ± ω L 2 are the energy eigenvalues of the left box.Now let us look at the operator P 1 .The operator acting on the left box is given by tracing out the degrees Using (2.55), we get which is precisely the same expression we had in (2.23).Therefore, the calculations in the previous section will go through -The K-complexity will grow linearly, following a quadratic scrambling phase.

Stitching Together Epochs
From (2.53), we can see that the late time linear growth rate of Krylov complexity during an epoch is given by the average energy of the left box, which we will denote by E L .Using E = 3 2 N k B T , we get Now, we will compute the time dependence of N L .Suppose the right box is sufficiently large for the gas to leak out completely.If the area of the hole is given by A, the leakage rate is given by [37] (2.61) We can immediately integrate the above equation if we assume that the temperature of the left box remains constant.This gives us where t L is the leakage time of the system.Treating N (t) as an instantaneous value, we can integrate (2.60) to obtain (2.63) Plotting the above function, we get figure 3. Krylov complexity keeps increasing and eventually levels off when t = t L .

Holographic Complexity of an Evaporating Black Hole
In this section, we will study the holographic complexity of a slowly evaporating black hole using the Complexity=Volume prescription [7,8].To make the calculations tractable, let us model the black hole by patching together a sequence of k static Schwarzschild spacetimes across negative energy null shock waves.The metric of the d + 1 dimensional black hole is then given by where We will choose f to have the following profile The mass of each patch is given by where For the patched-up spacetime to be a good approximation to an evaporating black hole, we will assume the width of each patch to be much smaller than the time scales at which the black hole mass changes considerably.Moreover, we will also assume that the width is larger than the scrambling time of the black hole.Each patch corresponds to a period where the black hole is effectively not evaporating.Therefore, we will adopt the terminology from our previous section and refer to each patch as an epoch.In particular, the patch between v = v i−1 and v = v i shock waves will be labeled as the i-th epoch.During an epoch, the black hole has a constant mass, and we can rewrite F (r, v) as F i (r).

Penrose Diagram
To understand the structure of the spacetime, it is instructive to draw its Penrose diagram.To simplify the discussion, we will restrict ourselves to 3 + 1-dimensions in this subsection.During an epoch, the tortoise coordinate is given by The corresponding outgoing Eddington-Finkelstein coordinates u i are defined as follows It is easy to see that u is discontinuous across the boundaries of the epochs.Therefore, if we go along a continuous curve, the coordinate u has a "jump" in its value as soon as we cross a shock wave.Consequently, employing identical u and v coordinates throughout all epochs would render continuous curves discontinuous in the corresponding Penrose diagram.This discontinuity is the cost we have to pay to keep the Penrose undeformed [38,39].During an epoch, the outgoing null Kruskal coordinate U i can be defined as follows [40]: Since v is globally defined, we can define the outgoing null Kruskal coordinate V everywhere as Using the definition of Kruskal coordinates, we can see that the horizon of the epochs will always be at U i = 0. Now let us figure out what happens to a continuous curve as it crosses a shock wave.We will denote the radial coordinate of the surface at the location of the shock wave by r S .Using (3.7) and (3.8), it is easy to verify that when r S is either in the interior or sufficiently far away from the horizon, we have the relation Now let us draw the Penrose diagram of the spacetime.We will use the same U and V coordinates across all the epochs.This gives us figure 4. A connected surface will be discontinuous in this diagram.We can locate the end points of these disconnected pieces by using (3.10).

Complexity=Volume
We can study the growth of complexity using the Complexity=Volume conjecture [7,8].Let us assume that there are only three epochs, with masses satisfying the relation M 1 > M 2 > M 3 ̸ = 0. We will see that extending our results to an arbitrary number of epochs is straightforward.The Penrose diagram of this spacetime is given in figure 4.
Consider spherically symmetric spacelike codimension-1 surfaces anchored onto a cutoff surface in the asymptotic region.We will assume that the cutoff surface is at r = r ∞ and the boundary anchoring time is denoted by t.The volume of this surface is given by Here Ω d−1 is the dimensionless area of the (d − 1)-dimensional unit sphere.Let us rewrite the volume integral as the following summation where V i is the volume of the portion of the surface located within the i-th epoch.We can then express each of these terms as follows: We can see that L i is independent of v. Therefore, there is a conserved quantity associated with the integral, which we will denote by E i : The volume integral is reparametrization invariant, allowing us to choose The maximal volumes are given by extremizing the action in (3.11).This gives us the following equations of motion: The part of the surface within an epoch can have a turning point where ṙ vanishes.We will denote this point by r i,min .We will see later that it is not necessary for the surface to have a turning point in an epoch.However, these turning points will be crucial in calculating late time growth rates during an epoch.
To characterize various features of the extremal volume surfaces, it is convenient to define an effective potential as follows: In particular, we can obtain the turning point r i,min from the zero of the effective potential Using the equations of motion, we can rewrite the volume integral as follows: The Complexity=Volume proposal suggests that the complexity of the black hole at time t is given by the volume of these extremal surfaces [8]: where ω i is the horizon radius during the i-th epoch.
First Epoch: Now let us look at the case where the boundary anchoring point is in the first epoch (Refer figure 5).This case reduces to the calculation of extremal volumes in a single-sided Schwarzschild black hole.Since we are in the first epoch, there will only be one term in the sum (3.19).The surface will always have a turning point.At late times, r 1,min approaches a constant radial surface in the interior of the black hole, located at the critical point of effective potential [8,41].We will refer to this surface as the accumulation surface.The location of this surface is given by (3.21) Since the width of each epoch is much larger than the scrambling time, the turning points will necessarily approach the accumulation surface at late times.Now let us calculate the growth rate of (3.11) as a function of the boundary anchoring time.From (3.16), we can see that Using (3.22), we can rewrite the volume integral (3.19) as follows: Taking a derivative w.r.t t and using Leibniz integral rule, we find the simple relation where we have used (3.18) to rewrite the energy in terms of the metric components.At late times, r 1,min approaches R 1,min , a constant.Using (3.21), we find that where Therefore, the late time complexity growth during the first epoch is given by Second epoch: Now let us look at the case where the boundary anchoring point is in the second epoch (see figure 6).Following the discussion in section 3.1, the extremal The surface will always have a turning point in the first epoch.Moreover, the turning point will be at the accumulation surface of the first epoch R 1,min .Three possibilities arise when we consider the turning point of the second epoch.The surface will not have a turning point during the very early stages.However, as boundary anchoring time increases, the surface will develop a turning point in the interior of the black hole.At late times, the turning point r 2,min will approach the accumulation surface of the epoch, given by

.29)
Now let us calculate the growth rate of these volumes.As in the previous section, we have Taking a time derivative w.r.t t, we get From (3.16), it is easy to see that the term in the second line is proportional to v1 (r S ) − v2 (r S ).Since v is continuous across each epoch, this function vanishes, and we get the simple expression: At late times, r 2,min approaches the accumulation surface (3.29).Substituting (3.29) in the above equation, we find the late time complexity growth during the second epoch to be given by where c d is defined in (3.26).i-th epoch: It is easy to extend the results to the i-th epoch.Complexity will undergo an initial transitional phase, after which it will settle to a linear growth characterized by the growth rate: might appear limiting.However, slow leakage is required if one wants to use intensive quantities like temperature at every instant of the process.The appearance of an ensemble-averaged semiclassical description in the context of black holes can be attributed to the existence of such quasi-static equilibriums [24].
Working out details, we saw that the Krylov complexity had the following behavior: • During an epoch, complexity goes through a scrambling phase and then transitions to linear growth.
• Complexity keeps increasing even as we cross the boundary of each epoch.However, the late time linear growth rate decreases with each successive epoch.
• When the gas has completely leaked out of the box, complexity levels off.
We claim that these results carry over to any chaotic quasi-static open quantum system if the operator under consideration satisfies ETH.Let us briefly outline this calculation by examining a system interacting with its environment.Consider an operator O which satisfies the ETH ansatz.The off-diagonal elements of this operator are given by where E i,j are the energy eigenstates of the system.R ij is a zero mean, unit variance random matrix whereas E and ω are given by (2.4).During an epoch, the Krylov complexity can be calculated using the Liouvillian of the system, following the assumptions in 2.3.1.If we assume F to decay as ω → ∞, then we can use the arguments in section 3.1 of [9] to see that the Krylov complexity undergoes a scrambling phase, followed by linear growth.As we have observed in the case of the slowly leaking gas, the linear growth rate will be proportional to the degrees of freedom of the system.Therefore, complexity will increase as we go from one epoch to the other, and it will eventually level off if all the degrees of freedom leak out of the system.This reproduces the advertised behavior.The holographic complexity, computed using a gravity calculation, displays the same behavior we described earlier.We can push this parallel further by comparing the late time growth rates (2.60) and (3.36).From the laws of black hole thermodynamics, we can see that the mass of the black hole plays the role of energy [43].Therefore, both calculations result in the same late time linear growth, provided we correctly identify the thermodynamic quantities on the black hole side.This provides further evidence to the claim that black holes can be described by a chaotic open quantum mechanical system with finite degrees of freedom when observed from the outside [44,45].Another manifestation of this proposal can be found in [24], where the entanglement entropy of the slowly leaking gas model matched the gravitational path integral calculation in [46].We can put these statements on a firmer footing by thinking of it as a consequence of black hole complementarity [44], which posits that the interior of a black hole can be thought to be described by a finite number of quantum mechanical degrees of freedom living on the stretched horizon of the black hole.The hard spheres in the left box assume the role of these degrees of freedom, while the particles in the right box model the outgoing Hawking radiation, allowing us to make a one-to-one map between an evaporating black hole and our two-box system.
Returning to the growth rate (3.36), we find that where S and T are the entropy and temperature of the black hole during that epoch.
Our results are in tandem with [47] and the 2d gravity calculation performed in [48].
One can also calculate complexity using the Complexity=Action (CA) prescription [49][50][51].However, to obtain meaningful results, the inclusion of a counterterm for null boundaries, similar to the one used in [38,39,52,53], might be necessary.

Figure 1 :
Figure 1: The figure on the left shows the growth of the Lanczos coefficients w.r.t n.There is an initial scrambling phase (shaded in green) where b E n grows as √ n, and then it saturates to a constant value.The figure on the right shows the corresponding Krylov Complexity growth.During the scrambling phase, the K-complexity grows quadratically.Then it transitions into a linear growth proportional to the average energy of the box.

Figure 2 :
Figure 2: Consider two cubic boxes in contact with each other.We fill the box on the left with N hard spheres.Let us assume that we are in the semi-classical limit where we can localize particles.At time t = 0, we poke a small hole on their common wall so that the gas leaks slowly into the right box.Let us denote the instantaneous number of particles in the boxes by N L and N R .

Figure 3 :
Figure 3: The figure shows the thermal Krylov complexity of a slowly leaking gas as a function of time.The complexity continues to rise, but eventually levels off.The inset figures show how K-complexity grows during two different epochs.As we move into future epochs, we observe a decrease in the late time growth rate.

Figure 4 :
Figure 4: Penrose diagram of the black hole spacetime when there are three epochs.We will assume the mass of each epoch to satisfy the relation M 1 > M 2 > M 3 ̸ = 0.The horizon is indicated by the 45-degree line.The shock waves are marked by the red lines.A spacelike surface, indicated by the teal color, will be disconnected in the diagram.The end points of these disconnected surfaces follow the ordering (3.10).

Figure 5 :
Figure 5: The left (right) figure shows the early (left) time behavior of the extremal volume surface when the boundary anchoring points are in the first epoch.We have not included the cutoff surface in the diagrams to avoid cluttering.The 45-degree line depicts the horizon.The grey dashed lines in the interior of the black hole correspond to the accumulation surfaces (Ref (3.21)) of the epochs.

Figure 6 :F 1 1 +F 2 2 ( 3
Figure 6: The left (right) figure shows the early (left) time behavior of the extremal volume surface when the boundary anchoring points are in the second epoch.The extremal surface is disconnected, and the endpoints of the surface are located using the relation (3.10).