In this work, we study approximations of supercritical or suction vortices in tornadic flows and their contribution to tornadogenesis and tornado maintenance using self-avoiding walks on a cubic lattice. We extend the previous work on turbulence by A. Chorin and collaborators to approximate the statistical equilibrium quantities of vortex filaments on a cubic lattice when both an energy and a statistical temperature are involved. Our results confirm that supercritical (smooth, “straight”) vortices have the highest average energy and correspond to negative temperatures in this model. The lowest-energy configurations are folded up and “balled up” to a great extent. The results support A. Chorin’s findings that, in the context of supercritical vortices in a tornadic flow, when such high-energy vortices stretch, they need to fold and transfer energy to the surrounding flow, contributing to tornado maintenance or leading to its genesis. The computations are performed using a Markov Chain Monte Carlo approach with a simple sampling algorithm using local transformations that allow the results to be reliable over a wide range of statistical temperatures, unlike the originally used pivot algorithm that only performs well near infinite temperatures. Efficient ways to compute entropy are discussed and show that a system with supercritical vortices will increase entropy by having these vortices fold and transfer their energy to the surrounding flow.
During the formation stage of a tornado, long narrow vortices often appear spontaneously in the region of tornado formation, then fold up and dissipate. These vortices also appear in other stages of the tornado’s existence. An example of such vortices in a strong tornado is shown schematically in
Recent progress in radar observations and numerical modeling of tornadogenesis and maintenance has revealed the importance of small-scale vorticity’s contributions to development of the large-scale rotation. For example,
Another study references [
multiple-Doppler-radar analyses, to the best of the authors’ knowledge, perhaps because they are too small scale to be detected (owing to the inherent smoothing during the analysis procedure) or because they occur only over a very shallow layer near the ground, where data collection can be difficult.”
The recent paper [
Many storm chaser videos show small-scale vortices entering tornadoes. Examples of small-scale vortices can be seen, for example, in the video in [
Supercritical or suction vortices also play an important role in tornadogenesis and tornado maintenance. A deeper discussion of this idea appears in [
intense multiple vortices within the larger flow is revealed by the radar data of the May 31, 2013, El Reno, OK tornado obtained by a Doppler on Wheels mobile radar [
Energy and entropy serve as fundamental descriptors of any thermodynamic system with applications in physics, atmospheric science, biochemistry, biophysics, and more. A recent example, reaching as far as DNA genetic mutations, is [
Atmospheric flows can be modeled using Euler’s equations relating velocity, pressure, and mass density of the fluid and external body forces. Typical flows are usually incompressible, so the divergence of the velocity field is zero. Isentropic flows are nearly incompressible when one of two scenarios occurs: either the flow speeds or the local changes in flow speeds along streamlines are small compared to the speed of sound in the medium [
With u being the fluid’s velocity, ρ its mass density, p its pressure, and b an external body force, all functions of position x ∈ ℝ 3 and time t ∈ ℝ , the governing equations for an incompressible fluid flow are
D u D t = − 1 ρ ∇ p + b , ∇ ⋅ u = 0 , D ρ D t = 0 , (1)
where D f D t = ∂ f ∂ t + ( u ⋅ ∇ ) f denotes the material derivative of a scalar function f . When applied to a vector function, the operator D D t applies component wise.
The vorticity field ξ of the velocity field u is given by the curl of u , i.e., ξ = ∇ × u . By applying curl to the first equation in (1) one can obtain an equation for ξ ,
∂ ξ ∂ t = ∇ × ( u × ξ ) + 1 ρ 2 ∇ ρ × ∇ p + ∇ × b .
In this equation, the first term in the right-hand side corresponds to the “barotropic” generation of vorticity (capturing the advection, stretching, and tilting of the vertical vorticity; see, e.g., [
In what follows, we will focus on flows in which vorticity is supported on long, narrow vortex tubes embedded in a larger irrotational flow. These may be baroclinic or barotropic in origin, and may demonstrate themselves, e.g., as the suction vortices discussed earlier [
Given a sufficiently fast decaying vorticity field ξ ( x ) in ℝ 3 , the system
ξ = ∇ × u , ∇ ⋅ u = 0
can be solved for the velocity field [
u ( x ) = − 1 4 π ∫ ℝ 3 ( x − x ′ ) × ξ ( x ′ ) | x − x ′ | 3 d x ′ .
This expression is known as the Biot-Savart law. From this, one has an expression for the kinetic energy of the flow [
E = 1 2 ∫ ℝ 3 ρ ( x ) | u ( x ) | 2 d x = 1 8 π ∬ ℝ 3 × ℝ 3 ρ ( x ) ξ ( x ) ⋅ ξ ( x ′ ) | x − x ′ | d x ′ d x + 1 8 π ∬ ℝ 3 × ℝ 3 ( ∇ ρ ( x ) × u ( x ) ) ⋅ ξ ( x ′ ) | x − x ′ | d x ′ d x .
For a homogeneous fluid the second term above vanishes and if density is rescaled so that ρ ( x ) ≡ 1 , the expression for the kinetic energy becomes [
E = 1 8 π ∬ ℝ 3 × ℝ 3 ξ ( x ) ⋅ ξ ( x ′ ) | x − x ′ | d x ′ d x , (2)
where we note, however, that the integral can be taken over supp ξ × supp ξ , so the integrals are evaluated over the vortex tubes only.
Following Chorin’s work, we now define an approximation to the energy (2) for an infinitely thin vortex filament, a “vortex line”. The assumption is that the vortex tube can be approximately divided into N narrow circular cylinders I i of equal length, for which we then have
E = 1 8 π ∬ ℝ 3 × ℝ 3 ξ ( x ) ⋅ ξ ( x ′ ) | x − x ′ | d x ′ d x = 1 8 π ∑ i ∑ j ≠ i ∫ I i ∫ I j ξ ( x ) ⋅ ξ ( x ′ ) | x − x ′ | d x ′ d x + 1 8 π ∑ i ∫ I i ∫ I i ξ ( x ) ⋅ ξ ( x ′ ) | x − x ′ | d x ′ d x ,
where the first term in the last expression corresponds to interactions of one of the cylinders with the others and is referred to as interaction or exchange energy, while the second term is referred to as self-energy and gives the contribution to the kinetic energy arising from interactions of nearby points along the vortex filament. In Chorin’s work, the self-energy term is neglected. For discussions of this term see, e.g., [
Finally, for an infinitely thin vortex line, the exchange energy is approximated in the following way. Assume that the vortex consists of N linear segments I i of equal length, whose midpoints are denoted by M i . Since vorticity along a vortex line is parallel to it, let ξ i denote the “vorticity” vector on the segment I i , which is in the direction of I i and has magnitude equal to the length of the segment I i . Then the exchange energy is approximated by
E = 1 8 π ∑ i ∑ j ≠ i ξ i ⋅ ξ j | M i − M j | . (3)
This scaling corresponds to the vortex having circulation equal to one [
To simplify the set of all possible configurations of a line vortex in 3D, we only consider those constrained to a cubic lattice ℤ 3 with lattice constant one [
| N | M, the number of SAWs of length N |
|---|---|
| 1 | 6 |
| 2 | 30 |
| 3 | 150 |
| 4 | 726 |
| 5 | 3534 |
| 6 | 16,926 |
| 7 | 81,390 |
| 8 | 387,966 |
| 9 | 1,853,886 |
| … | … |
| 36 | 2,941,370,856,334,701,726,560,670 ≈ 2.9 × 1024 |
| 1000 | ≈1.5 × 10671 |
In this paper we demonstrate an improvement of the computation of average energy and entropy compared to previous attempts, and thus fairly small values of N (≤1000) will be considered. These are small values compared to the computational state of the art, however the numbers of distinct SAWs, M, are still large, ranging from M ≈ 10 67 for N = 100 to M ≈ 10 671 for N = 1000 [
In the next sections we describe the relevant statistical mechanics and the algorithm used to obtain accurate results.
We now briefly review relevant concepts from statistical mechanics. Given a positive integer N corresponding to the number of segments in a self-avoiding walk on a cubic lattice, we denote by M the total number of possible SAWs starting at the origin. We denote by x i , i = 1 , ⋯ , M , the possible SAWs, or vortex configurations of length N. The vortex configurations x i represent individual states in the system S N = { x i : i = 1, ⋯ , M } .
The energy of each configuration, a state energy, will be denoted by E i = E ( x i ) with E defined in (3). As introduced in [
p i = e − β E i Z , Z = ∑ i = 1 M e − β E i , (4)
where Z is called the (canonical) partition function and β = 1 k B T is sometimes
referred to as “coldness” as it is proportional to the reciprocal of the (statistical) temperature, T. The Boltzmann constant, kB, will be assumed equal to 1 in subsequent computations but we will use it in the general results in this section. Note that when β = 0 all states are equally likely and p i = 1 / M for i = 1 , ⋯ , M . This is sometimes referred to as the polymeric case as equally likely configurations are considered when modeling the behavior of polymers [
Notice that the relationship β = 1 k B T can be also used to define the temperature T = 1 k B β . Consider β increasing from −∞ to +∞. When β increases from
−∞ to 0, this corresponds to T decreasing from 0 to −∞; when β increases from 0 to +∞, this corresponds to T decreasing from +∞ to 0. Put together, when one considers the temperature T, temperature first increases from T = 0 to T = + ∞ ; considering the one-sided limits as β → 0 , one can identify T = + ∞ with T = − ∞ and simply write T = ∞ . Then temperature further increases as T changes from T = ∞ through negative values towards 0. Thus negative temperatures are higher than positive temperatures in this sense. (This idea corresponds to traversing a circle obtained by transforming the real number line into a circle by identifying +∞ and −∞, both for β and T; this identification is a special case of the Möbius transformation of the complex plane.) Another explanation for why negative temperatures are higher than positive ones is based on energy (and entropy) considerations. For further details, see [
Using the Boltzmann distribution (4), we can now define in a standard way the average of any function of the states x i ∈ S N for any finite β. For example, the average energy 〈 E 〉 of the system S N at a given β (or temperature T) is given by
〈 E 〉 = ∑ i = 1 M p i E i . (5)
One of the goals of this paper is to reliably approximate 〈 E 〉 for any finite β, which would extend the results of Chorin (see, e.g., [
The Gibbs entropy of the system S N at a given β is given by
S = − k B ∑ i = 1 M p i log p i , (6)
which can also be viewed as an average quantity, since one can write
S = ∑ i = 1 M p i ( − k B log p i ) = ∑ i = 1 M p i S i
and view S i = − k B log p i as an “entropy” of the state x i . Also note that if β = 0 , then, since p i = 1 / M , we have S = k B log M , which corresponds to Boltzmann’s definition of entropy.
Taking a partial derivative of the logarithm of the partition function in (4) with respect to β , we obtain
− ∂ log Z ∂ β = 1 Z ∑ i = 1 M E i e − β E i = 〈 E 〉 . (7)
Using (4), expression (6) can be written as
S = ∑ i = 1 M p i ( k B β E i + k B log Z ) = k B β 〈 E 〉 + k B log Z ,
and differentiating it with respect to 〈 E 〉 and using (7) gives the well-known result
∂ S ∂ 〈 E 〉 = k B ∂ β ∂ 〈 E 〉 〈 E 〉 + k B β + k B ∂ log Z ∂ β ∂ β ∂ 〈 E 〉 = k B β = 1 T . (8)
Finally, the Helmholtz free energy of the system is
F = 〈 E 〉 − T S = ∑ i = 1 M p i ( E i + k B T log p i ) ,
where, in view of (4) and the relationship between β and T, the term in parentheses equals − k B T log Z , i.e., a constant independent of i, and hence the expression for the Helmholtz free energy gives
F = 〈 E 〉 − T S = E i + k B T log p i for any i = 1, ⋯ , M . (9)
In particular, this means that the entropy (and also the Helmholtz free energy and the partition function) can be computed from β, 〈 E 〉 , E i , and p i for any i provided these values are known since (9) can be rewritten as
S = k B ( β 〈 E 〉 − β E i − log p i ) for any i = 1, ⋯ , M . (10)
Equations (8) and (10) will be used later in Section 7, where we discuss approximating the entropy of the vortex filament system.
In order to reliably compute various statistical quantities of interest, such as the average energy (5) or the entropy (6), one usually employs Markov chain Monte Carlo (MCMC) techniques, since the space S N of possible configurations of the vortex filament of length N is too large for direct enumerations. As mentioned in Section 3, complete enumerations have been achieved only for small values of N up to date [
Here we briefly review the Metropolis-Hastings algorithm that we use in this work. We will consider two configurations, x and x’, their energies E and E’ given by (3), and their probabilities P ( x ) and P ( x ′ ) given by (4). We define g ( x ′ | x ) to be the probability of proposing a transition from configuration x to configuration x’; A ( x ′ | x ) to be the probability of accepting a transition from x to x’; and P ( x ′ | x ) to be the probability of transitioning from x to x'. Note that P ( x ′ | x ) = A ( x ′ | x ) g ( x ′ | x ) and that we can easily compute the ratio P ( x ′ ) / P ( x ) = e − β ( E ′ − E ) = e − β Δ E since the difficult-to-compute partition function Z cancels out.
The detailed balance condition states that at equilibrium, each elementary process is in equilibrium with its reverse process, so we require
P ( x ′ | x ) P ( x ) = P ( x | x ′ ) P ( x ′ ) .
Since P ( x ′ | x ) = A ( x ′ | x ) g ( x ′ | x ) , the detailed balance condition can be rewritten as
A ( x ′ | x ) A ( x | x ′ ) = P ( x ′ ) P ( x ) ⋅ g ( x | x ′ ) g ( x ′ | x ) = e − β Δ E ⋅ g ( x | x ′ ) g ( x ′ | x ) .
A typical way to satisfy this condition is to choose
A ( x ′ | x ) = min { 1, e − β Δ E ⋅ g ( x | x ′ ) g ( x ′ | x ) } , (11)
and an alternative choice is [
A ( x ′ | x ) = 1 1 + e β Δ E ⋅ g ( x | x ′ ) g ( x ′ | x ) . (12)
Note that sometimes g ( x ′ | x ) ≠ g ( x | x ′ ) and the ratio of the proposal probabilities has to be computed and retained in the algorithm. Either of the two acceptance probabilities (11) or (12) is then compared to a random number picked uniformly from ( 0,1 ) and based on the comparison the proposed transition is accepted or not. In this way one is able to generate a sequence of configurations { x 0 , x 1 , ⋯ } ⊂ S N in which each consecutive configuration is either the one resulting from an accepted transition, or a repeat of the previous configuration if the transition was rejected. The sequence can then be used to approximate the underlying probability distribution and its properties, and a good set of possible transitions is essential for fast and reliable performance.
Another requirement for the MCMC algorithm to work properly in sampling the configuration space is that it is ergodic, i.e., it is able to propose a finite sequence of transitions from any configuration x 1 ⊂ S N to any other configuration x 2 ⊂ S N .
An algorithm for proposing new configurations from old ones used in the previous works [
In the context of our problem the pivot algorithm will not perform well in many scenarios due to the involvement of the vortex filament energy (3) in (4) and (5) and due to the fact that the pivot transformations are not local—in any given step all of the points after the pivot are expected to change position leading to a possibly large change in the filament energy. As commented in [
Various modifications of the pivot algorithm have been considered in the literature for various reasons. For example, additional possible moves useful in modeling polymers are discussed in [
The rigidity of the pivot algorithm for problems that involve energy and the Boltzmann distribution away from the polymeric case ( β = 0 or T = ∞ ) implies that a different approach is needed. In cases with | β | ≫ 0 the transformations need to be local enough so that much of the shape of the current SAW remains the same, thus preserving much of the SAWs energy, and only a small part should be changed.
We have found that a very simple algorithm that we describe below works well for this problem across a wide spectrum of the values of β. The algorithm will use two types of transformations: one designed specifically to help with straightening out non-straight configurations at negative values of β, and one designed to help with compressing configurations in order to lower their energy at positive values of β. We next describe these two types of transformations and later we comment on their relationship to others existing in the literature.
For large (in absolute value) negative values of β, the pivot algorithm is very unlikely to straighten out a configuration with all but one segment in one direction (or similar, more complicated configurations), since in order to remove the “kink,” the whole part after the kink has to be temporarily transformed as well, resulting in a significant change in energy. An easy solution is to simply modify the one non-aligned segment while keeping the directions of the other segments the same. This still modifies the positions of the points after the kink, but it results in a small change in the filament energy because the relative positions of the points stay the same. We generalize this idea to the following algorithm for an N-step SAW:
First Reconstruction Algorithm:
1) Randomly select a number n_steps between 1 and N of consecutive steps to be modified.
2) Randomly select a subwalk of length n_steps to be modified.
3) Randomly reconstruct the selected subwalk.
4) Connect the reconstructed subwalk to the remaining one or two pieces by translation.
This algorithm is clearly ergodic since the reconstruction can be done on the whole SAW. Inverses of transformations are included and selected with equal probabilities, so the algorithm satisfies the detailed balance condition as well when the usual Monte Carlo acceptance probability is used. Practical considerations here include the choice of probability distributions in steps 1) and 2) (we used an exponential distribution in 1) to emphasize shorter reconstructions and a uniform distribution in step 2) and how much care should go into step 3) (e.g., random reconstruction vs. constructing a true SAW; we used a random reconstruction). It is easy to see that the configurations with one misaligned segment are easy to transform into a straight configuration with little energy change.
For large positive values of β, the filament configurations are expected to have small energy and be folded up and “balled up” to some degree in order to lower their energy. The first reconstruction algorithm is not an ideal candidate here due to the fact that the part of the filament after the reconstructed subwalk is likely to be translated during the reconnection, likely leading to self-intersections of already balled up configurations. Motivated by transformations proposed for configurations with fixed endpoints [
Second Reconstruction Algorithm:
1) Choose a number K between 2 and N for maximum number of allowed reconstructed steps.
2) Randomly select a number n_steps between 2 and K of consecutive steps to be modified.
3) Randomly select a subwalk of length n_steps to be modified.
4) Permute the selected steps to generate a new subwalk.
5) Connect the reconstructed subwalk to the remaining one or two pieces.
Combining the two reconstruction algorithms results in an algorithm that is ergodic due to the ergodicity of the first algorithm. The second algorithm also clearly satisfies the detailed balance condition. Practical considerations here include the following. The number K can be chosen arbitrarily, but we found that K ≈ N 2 / 3 not only roughly corresponds to the number of surface points in a perfectly “balled up” configuration, it also appears to lead to reasonably small minimum energy values. For the probability distributions in 1) and 2), we again used an exponential and a uniform distribution, respectively. The permutation in 3) could be a random permutation; we found that choosing a (uniformly) random cyclic permutation led to good results. In general, even though we never attempted to directly compute minimum energies, we found that our lowest-energy configurations in the range − 100 ≤ β ≤ 100 had energies below or within a fraction of a percent of the predicted minimum values as shown in Section 7.5.
We note that the first algorithm includes many of the standard simple transformations mentioned earlier, like the end flips, corner flips, and crankshaft moves. Some of the pull moves from [
It is worth pointing out that an alternative approach to our work exists and consists in approximating the density of states using Wang-Landau sampling [
In this section we present numerical results for the computation of the average energy 〈 E 〉 and the entropy S as a function of the length of the vortex filament, N, and the inverse temperature, β. These results extend the results presented in [
In the individual subsections we present and discuss the validation of the algorithm in cases where the exact results are known, results for the average energy (5) for various lengths of the filaments, we also discuss and compute the entropy of the system, and we present some observations related to the minimum energy configurations, even though that is not the main focus of this paper.
Without knowing the exact values of the average energy (5), it is impossible to assess the accuracy of the computed results. Due to the growth rate of the number of SAWs as a function of the number of steps N (see
To demonstrate the capabilities of the MCMC algorithm, we perform the following test. Given N and β, start with a straight configuration but first perform 10N iterations with β = 0 to get away from it. Next, first perform 100,000 iterations with the given β to forget history and then perform 200,000 averaging iterations to approximate 〈 E 〉 for that β. Repeat with various values of β in the interval of interest. The computed results, together with the exact values, are shown in
In this section, we present the numerical results for computing the average energy, 〈 E 〉 , given in (5), using the proposed algorithm with localized transformations that comprises of the first and second reconstruction algorithms described in the previous section.
The implementation details are as follows. As in [
E max ( N ) = 1 4 π [ N ∑ k = 1 N − 1 1 k − ( N − 1 ) ] . (13)
In light of (4), the straight configurations are the most likely configurations for negative temperatures ( β < 0 ), and we expect 〈 E 〉 ≈ E max when β is large (in absolute value) negative. Therefore, we start our computation at one such β ( β first = − 20 for the results below) and compute an approximation of the average energy for this value of β using the straight configuration as a starting point. We then increase β by a small increment ( Δ β = 0.5 in the results below), use the last configuration from the computation with the previous value of β as a starting point, and compute 〈 E 〉 with the new β. We continue this way until a stopping value of β has been achieved ( β last = 100 in the results below).
The average energy for each value of β is computed by first allowing for a number of “burn-in” transformations to forget recent history and then by performing a different number of averaging transformations to compute 〈 E 〉 . The numbers of iterations depend on N and β and vary between 10,000 and 55,000 for burn-in and between 500,000 and 1,500,000 for averaging.
During each averaging iteration, either the first or the second reconstruction algorithm is selected randomly with probability 1/2, which could conceivably be modified based on the value of β. For the first reconstruction algorithm a ratio r 1 ∈ ( 0,1 ] is passed in and a geometric probability distribution with the ratio r 1 is constructed for the numbers 1 through N. (If r 1 = 1 , the distribution is uniform.) A random number n_steps is drawn from the set { 1, ⋯ , N } according to this probability distribution. Clearly, for r 1 < 1 it is more likely to reconstruct shorter subwalks than longer ones. Finally, a starting point of the subwalk to be reconstructed is chosen randomly uniformly from possible candidates. For the second reconstruction algorithm, a similar process is followed. With a ratio r 2 ∈ ( 0,1 ] , first a geometric probability distribution with the ratio r 2 is constructed for the numbers 2 through K, where K is the integer nearest to N2/3. Then a random number n_steps is drawn from the set { 2, ⋯ , K } according to this probability distribution. Finally, a starting point of the subwalk to be reconstructed is chosen randomly uniformly from possible candidates. At this point, a reconstruction of the subwalk is performed according to steps 3) - 4) in the relevant reconstruction algorithm and the new filament is tested for self-avoidance and for acceptance in the MCMC algorithm. If it is self-avoiding and accepted using the Boltzmann probability criterion, the newly constructed filament becomes the new filament in the Markov chain; otherwise the current one is repeated.
In
when β < 0 and is large in absolute value. Specifically, for β ∈ [ − 20, − 10 ) the average energies are near their maximum values (13), corresponding to, on average, straight configurations. As temperature is lowered in the computation, average energy decreases as expected in light of (4). Due to the finite lattice spacing, for each N there is a minimum energy a filament of N steps can achieve, so it is not surprising that the average energies level off as β → ∞ . This is demonstrated in
An important point to be emphasized here is that the vortices start folding when β (and the temperature) is still negative. Note first that for β < 0 the straight configuration has the largest probability (4) of occurring, but as β approaches 0, the probability density is becoming more and more uniform. Therefore, the average energy decreases as seen in
Each displayed result is a mean of 6 configurations that are within 1% of the mean energy for each β. The error bars show ±1 standard deviation of the 6 results. We can see how the average lengths of the filaments decrease as β → 0 − , corresponding to the folding of the filaments.
In Chorin’s work [
In order to validate these results and extend them to a larger range of β, we used as benchmarks the cases for which we had the complete enumerations ( N = 3 , ⋯ , 9 ). The exact entropies for these cases are shown in
M ≈ M ˜ = A μ N N θ ( 1 + c N − Δ + k ( − 1 ) N N − α ) , (14)
where A = 1.1951966888 , μ = 4.6840041570 , θ = 0.1597395125 , c = 0.1227360755 , Δ = 1.4315024046 , k = − 0.0619076482 , and α = 1.8985141134 . We note that the relative errors resulting from this formula for N = 14, ⋯ ,36 are all below 2 × 10−6; this results in absolute errors for the entropy, log M ˜ − log M , to be of the same order. Consequently, the resulting approximation
S / N = ( log M ) / N ≈ ( log M ˜ ) / N (15)
is expected to be quite good and perhaps improve for longer filaments due to the N in the denominator.
Using (14) and (15) to approximate S/N for N = 351 , we obtain S / N ≈ 1.54733 , which appears significantly below the maximum value of about 1.67 estimated from
Improvements of the Meirovitch algorithm have appeared in the literature more recently and have been applied to magnetic systems, polymers, peptides, and liquids (see, e.g., [
The approximation to p ( x ) , denoted by p HSMC ( x ) , is computed as a product of conditional “transition” probabilities that the kth step of the filament is in the given position as in x, given that the previous k − 1 steps are fixed as in x. To generate these probabilities, we would keep the first k − 1 steps fixed and apply our algorithm as used to compute 〈 E 〉 by allowing reconstructions only between the nodes k through N. The transition probability would then be approximated by the ratio of the filaments that agree with the kth step of x versus all filaments generated in the Monte Carlo run. Clearly, this process would be computationally intensive, since N − 1 Monte Carlo runs (progressively shorter and shorter) would have to be performed to compute a single p HSMC ( x ) . On the other hand, p ( x ) could be approximated arbitrarily closely by allowing sufficiently long simulations if a good sampling algorithm is used.
With the computed p HSMC ( x ) the Helmholtz free energy, F, could be approximated via (9),
F ≈ F HSMC = E ( x ) + 1 β log p HSMC ( x ) ,
where E ( x ) is the easily computed energy of the filament x. Subsequently, the entropy could be approximated through the relationship F = 〈 E 〉 − T S if the average energy has been computed, as shown in (10),
S ≈ S HSMC = k B β ( 〈 E 〉 − F HSMC ) = k B ( β 〈 E 〉 − β E ( x ) − log p HSMC ( x ) ) .
It follows that the absolute error in the entropy computation, | S − S HSMC | , will be affected both by the accuracy of the HSMC computation of p HSMC ( x ) as well as the accuracy of the computed 〈 E 〉 ; more precisely, it will be roughly bounded by the sum of the relative error in approximating p ( x ) and | β | times the absolute error in approximating 〈 E 〉 (all multiplied by k B ).
Note that to compute the entropy of the system of filaments of length N for many possible values of β (such as shown in
A very efficient alternative to computing the entropy can be used if good approximations to 〈 E 〉 have been computed across an interval of β values and a single value for entropy is known. Recall the relationship (8), restated here for convenience:
∂ S ∂ 〈 E 〉 = k B β . (16)
If the β interval is partitioned with values β i , the corresponding (average) energies are denoted by E i , and the (sought) entropies by S i , then, provided S is smooth enough, (16) can be discretized as
S i + 1 − S i E i + 1 − E i = k B β i + 1 / 2 + ( Δ β i ) 3 48 ∂ 3 S ∂ 〈 E 〉 3 | 〈 E 〉 = e ˜
for some e ˜ between E i and E i + 1 , where β i + 1 / 2 = ( β i + 1 + β i ) / 2 and Δ β i = | β i + 1 − β i | . This then leads to the following Euler-like method for approximating the entropy,
S i + 1 = S i + k B β i + 1 / 2 ( E i + 1 − E i ) , (17)
provided at least one of the values S i is known to start the recursion. Since, as discussed above, the entropy when β = 0 can be computed for small N or approximated for larger N by k B log M ˜ using (14), a starting value is readily available. Alternatively, one can use the HSMC algorithm to approximate the entropy for any one value of β, for example β = 0 .
To validate this approach, we first apply it to the cases with N = 3 , ⋯ , 9 and with exact (average) energy values provided at equidistant values of β with Δ β = 0.5 . The results are shown in
For N = 100 , 200 , ⋯ , 1000 , we use (14) and (15) to get an approximation for the entropy at β = 0 , and apply algorithm (17) with the precomputed average energy values. For the values of 〈 E 〉 shown in
fact that S/N appears to be approaching a finite limit suggests that the number of energy minimizing configurations grows exponentially as N → ∞ or that an exponentially growing number of configurations can be found in the vicinity of the minimum-energy configurations.
To make a comparison to the results in [
Even though we do not have a direct comparison to exact values of the average energy (5) for large values of N, it appears that our algorithm allows for accurate approximation of the average energies for negative values of β and for values close to 0. How accurate the computed energies for large positive values of β are is less clear. It follows from (4) that for such values the probabilistically preferred configurations are those with lowest energies, but it is not clear what the lowest possible energy is. This is in contrast with the largest possible energy value that is clearly associated with the straight filament and its value is easily computable via (13).
Intuitively, the energy expression (3) suggests that the lowest energy will be associated with a filament folded in such a way that individual segments line up close to each other in an antiparallel way, resulting in large negative contributions to the energy, and also in such a way that there are many right angles
which contribute zero energy. From a fluid mechanics point of view this is also intuitive since antiparallel vortices with the same circulation should contribute negligible kinetic energy away from the vortices.
Even though this isn’t one of the goals of this paper, to assess how well our algorithm with localized transformations can reconstruct the low-energy configurations, we first computed the lowest possible energies for short filaments. We fully enumerated all filaments of lengths up to N = 18 , and then based on these results we made simplifying assumptions to extend the results up to N = 21 . These results (with 6 significant digits) are shown in
Looking at the limited set of data, it appears to exhibit a strong linear pattern. The best fit line has the equation E min ~ ( − 0.604671 N + 1.13186 ) / ( 4 π ) and R 2 = 0.996184 . We then used our Monte Carlo algorithm and attempted to locate the minimum-energy configurations for larger values of N. Since these have not been validated by another approach, we will not report them here. These computed energies follow the linear trend and when using them to generate a best-fit line, we obtain the equation
E min ~ 1 4 π ( − 0.612418 N + 1.19011 ) (18)
with R 2 = 0.998585 . Using this slightly steeper line to generate predictions for the minimum energies for N = 100, ⋯ ,1000 , we obtain the results in the column labeled “Predicted E min ” in
| N | 4 π × E min ( N ) |
|---|---|
| 2 | 0 |
| 3, 4 | −1 |
| 5 | −2 |
| 6 | −2.29289 |
| 7 | −3.58579 |
| 8 | −3.45603 |
| 9 | −4.32627 |
| 10 | −4.86986 |
| 11 | −5.80649 |
| 12 | −5.86948 |
| 13 | −6.94895 |
| 14 | −7.14629 |
| 15 | −8.10289 |
| 16 | −8.37101 |
| 17 | −9.17890 |
| 18 | −9.63666 |
| 19 | −10.6340 |
| 20 | −10.9049 |
| 21 | −11.7208 |
| N | Predicted E min | Best found E min | % of E min |
|---|---|---|---|
| 100 | −4.77876 | −4.97778 | 104.16 |
| 200 | −9.65223 | −9.76269 | 101.14 |
| 300 | −14.5257 | −14.8185 | 102.02 |
| 400 | −19.3992 | −19.4220 | 100.12 |
| 500 | −24.2726 | −24.4001 | 100.53 |
| 600 | −29.1461 | −29.6006 | 101.56 |
| 700 | −34.0196 | −34.2256 | 100.61 |
| 800 | −38.8930 | −38.9554 | 100.16 |
| 900 | −43.7665 | −43.7911 | 100.06 |
| 1000 | −48.6400 | −48.3705 | 99.45 |
To illustrate the idea of low-energy configurations being folded and compressed, we show in
It is also interesting to point out that all the lowest-energy configurations, whether exact or found approximately in our simulations, ended up with their endpoints in the same unit square on the cubic lattice as illustrated in
our numerical algorithm. Although not presented here, some preliminary work has also been done on considering these configurations as subwalks for self-avoiding polygons.
In this paper we explore vortices using a statistical mechanical approach. We consider vortices in negative, infinite, and positive temperature states, a concept previously studied in [
Our results confirm that the supercritical vortices (modeled by straight filaments) correspond to the highest kinetic energy of the flow and also correspond to negative temperatures in this model. The lowest-energy configurations are folded up and compressed to a great extent. From the point of view of the flow of energy, the negative-temperature, high-energy supercritical vortices are expected to lose (some of) their energy to the surrounding flow, which can now be understood as a natural process of increasing the entropy of the system. In [
Our results are not intended to simulate the dynamics of supercritical vortices, only their corresponding statistical equilibrium energies. From this point of view, the results show that a more folded vortex corresponds to a flow with lower kinetic energy. Hence, if a vortex of a fixed length folds, the flow would lose some of its kinetic energy, and this loss would have to be interpreted as a transfer of the energy to the surrounding flow.
We also see that since entropy is highest for β = 0 ( T = ± ∞ ), a system containing a supercritical vortex, which corresponds to high negative temperatures, would be naturally driven towards the state where β = 0 and thus lower the energy associated with the vortex. This corresponds to the vortex folding and energy being transferred to the surrounding flow.
In this work we proposed and utilized a simple algorithm for the computation of statistical equilibrium quantities on a cubic lattice when both an energy and a statistical temperature are involved. This was a consequence of the fact that the pivot algorithm used in [
At the other end of the inverse temperature spectrum, when the values of β are large positive, the Boltzmann probability distribution strongly favors lowest-energy configurations, very much folded up and compressed into a small volume. While the maximum-energy configurations are obviously the straight configurations with their energies easily computed for any N, the exact minimum-energy configurations are not known to us, nor are the actual minimum energies. However, the pivot algorithm MCMC simulation struggles to get anywhere close to the hypothesized minimum values, thus begging for a different approach to the problem. Some possibilities might include a statistical approach studied, for example, in [
The algorithm utilized in this work seems to perform well for all possible temperature values. It uses two types of transformations: one designed specifically to help with straightening out non-straight configurations in order to increase their energy at negative values of β, and one designed to help with compressing configurations in order to lower their energy at positive values of β. The latter was motivated by previous work for generating self-avoiding walks with fixed endpoints [
Having reliably approximated the values of equilibrium energy, we also proposed an efficient way to compute equilibrium entropy for all temperature values using an algorithm that mimics the midpoint Euler’s method for numerically approximating solutions of differential equations in that it approximates the solution to ∂ S / ∂ 〈 E 〉 = k B β instead of computing the entropy directly. The algorithm requires the knowledge of 〈 E 〉 for a range of values of β, as well as an initial value of the entropy S for one value of β. This value (for β = 0 ) was readily estimated using existing works on approximate number of possible N-step self-avoiding walks on a cubic lattice. As far as we are aware, we have not seen this approach used in any previous work.
In this work we have not addressed some of the other concepts approximated in the earlier works of Chorin et al. [
Sincere thanks to the Office of Undergraduate Research and Graduate Opportunity (URGO) at Augsburg University and to Dean and Amy Sundquist for providing the research opportunity and financial support for Eric Bibelnieks, Robert Laskowski, and Alek Lukanen. The authors would also like to thank Arkady Shemyakin and the anonymous referees for their useful feedback and suggestions.
The authors declare no conflicts of interest regarding the publication of this paper.
Bělík, P., Dokken, D.P., Shvartsman, M.M., Bibelnieks, E., Laskowski, R. and Lukanen, A. (2023) Equilibrium Energy and Entropy of Vortex Filaments in the Context of Tornadogenesis and Tornadic Flows. Open Journal of Fluid Dynamics, 13, 144-176. https://doi.org/10.4236/ojfd.2023.133012