White privilege is an absence of the consequences of racism. An absence of structural discrimination, an absence of your race being viewed as a problem first and foremost.

― Reni Eddo-Lodge

Every Monday I stare at a blank page and write. Sometimes it is obvious with work demands driving my thoughts in a very focused direction. I’d love to be there right now.  Work is pissing me off, and providing no such inspiration, only anger. It’s especially unfortunate because I ought to have something to say on differential equations. Last week’s post was interesting, and the feedback has provided wonderful topics to pick up. One comment focused on other constraints and invariants for modeling that equations commonly violate. A wonderful manager I had from Los Alamos argued with me about nonlinearity, and integral laws with respect to hyperbolic PDEs and causality. Both of these topics are worthy of deep exploration, but my head isn’t there. The issue of societal inclusivity and the maintenance of power structures looms in my thinking. The acceptance of individuality and personal differences within our social structures is another theme that resonates. So, I take a pause from technical to ponder the ability of our modern World to honor and harness the life work and experience from others especially those who are different. The gulf between our leader’s rhetoric and actions is vast leading to poor results, and maintenance of traditional power.

Why am I concerned?

It does strange things to you to realize that the conservative establishment is forcing you to be a progressive liberal fighter for universal rights.

― Brandon Sanderson

This whole topic might seem to be odd coming from me. I am a middle aged white manwho is a member of the intellectual elite. I’m not a racial minority, a foreigner or part of the LBGTQ community although I have a child who is. Of course, my (true) identity is far more complex than that. For example, my personality an outlier at work. It makes many of my co-workers extremely uncomfortable. I’m an atheist and this makes people of faith uncomfortable. Other aspects of my life remain hidden too, maybe they aren’t germane; maybe they simply would make people uncomfortable; maybe they would result in implicit shunning and rejection. All of this runs against the theme of inclusion and diversity, which society as a whole and our institutions play lip service to. In the backdrop of this lip service are entrenched behaviors, practices and power that undermines inclusion at every turn. Many forms of diversity have no protection at all, and the power structures attack them without relief. The talk is all about diversity and inclusion, the actions are all about exclusivity. Without genuine leadership and commitment diversity and inclusion fail. Worse yet, a great deal of our leadership has moved to explicit attacks on diversity as a social ill.

A great democracy has got to be progressive or it will soon cease to be great or a democracy.

― Theodore Roosevelt

Let’s pull this thread a bit with respect to the obvious diversity that we see highlighted societally, race, gender, or sexual identity. These groups are protected legally and the focus of most of our diversity efforts. The discrimination against these groups of people has been pervasive and chronic. Moreover, there are broad swaths of our society that seek to push back progress and reinstitute outright discrimination. The efforts resisting diversity are active and vigorous. In lieu of the explicit and obvious discrimination we have plenty of implicit institutional discrimination and oppression. The most obvious societal form that implicitly targets minorities is the drug war and the plague of mass incarceration. It is a way of instituting “Jim Crow” laws within a modern society. Any examination of the prisons in the United States shows massive racial inequity. Police enforce laws in a selective fashion letting whites go, while imprisoning blacks and Hispanics for the same offenses. Prison sentences are also harsher toward minorities. Other minority groups have been targeted and selected for similar treatments. In these cases, everyone is claimed to be under the same legal system, but the execution of the law lacks uniformity. At work we have policies and practices often revolving around hiring that have the same effect. They work to implicitly exclude people and undermine the actual impact of diversity.

Equality before the law is probably forever unattainable. It is a noble ideal, but it can never be realized, for what men value in this world is not rights but privileges.

― H.L. Mencken

Other practices hurt women in an implicit manner. These practices are relabeled as “traditional values” and are reflected through health care and sexual harassment. The implicit message is that women should be barefoot and pregnant, second class citizens whose role in society is child rearing and home making. Young women are also there to provide sexual outlets to men, although those that do are sluts. They are not worthy of enjoying the same sexual pleasure the men are free to pursue. The whole of the behavior is focused on propping up men and the center of society, work and power. None of their needs or contributions in the realm where men traditionally rule is welcome. Our health care coverage and laws completely reflect this polarity where women’s needs are controversial. Men’s needs are standard. Nothing says this more than the attitudes toward reproductive and sexual health. Men are cared for; women are denied. The United States is a nexus of backwards thought with labor laws that penalize women in the name of the almighty dollar using religious freedom as an excuse.

If we cannot end now our differences, at least we can help make the world safe for diversity.

― John F. Kennedy

In the past decades we have seen huge strides for the LBGTQ community. The greatest victory is marriage equality. Discrimination is still something many people in society would love to practice. At an implicit level they do. Everywhere that discrimination can be gotten away with, it happens. Many people remain in the closet and do not feel free letting people know who they really are. Of course, this is a bigger issue for transgender people for whom hiding is almost impossible. At the same time the level of discrimination is still largely approved by society. Most of the discrimination is the result of people’s innate discomfort with LBGTQ sexuality and opening themselves to even considering being somewhere on any non-standard spectrum of sexuality. It becomes an issue and a thought they would just as soon submerge. This discomfort extends to other forms of non-standard expressions of sexuality that invariably leak out into people’s everyday lives. This sort of discomfort is greater in the United States where sexuality is heavily suppressed and viewed as morally inferior. Our sexuality is an important part of our self-identity. For many people it is an aspect of self that must be hidden lest they suffer reprisals.

This is a global issue

President Donald Trump reacts before speaking at a rally at the Phoenix Convention Center, Tuesday, Aug. 22, 2017, in Phoenix. (AP Photo/Alex Brandon)

Hate and oppression are rising Worldwide. This hate is a reaction to many changes in society. Ethnic changes, migrations, displacement and demographics are undermining traditional majorities. Global economics and broad information technology/telecommunication are also stressing traditional structures at every turn. Disparate people and communities can now form through the power of this medium. At the same time disinformation, propaganda and wholesale manipulation are empowering the masses to both good and evil. Some of these online communities are wonderful such as LGBTQ people who can form greater bonds, and outreach to similar people. It also allows hate to flourish and bond in the same way. The technology isn’t bad or good, and its impacts are similarly divided. It is immensely destabilizing. Traditional culture and society is also rising up to push back. The change makes people uncomfortable. Exclusion and casting the different out is a way for them to push back. They respond with a defense of traditional values and judgements grounded in exclusion of those who don’t fit in. As with most bigotry and exclusion, it is fear based. Fear is something promoted across the globe by the enemies of inclusion and progress. The same fear is being harnessed by the traditional powers to solidify their hold over their hegemony. This fear is particularly acute with the older part of the population who also tend to be more politically active. These two things form the most active implicit threat to achieving diversity.

In many cases this whole issue is framed in terms of religion. Many traditional religious views are really excuses and justification for exclusion and bigotry. The aspects of the religious traditions focused on love, compassion and inclusion are diminished or ignored. This sort of perversion of religious views is a common practice by authoritarian regimes who harness the fear-based aspects of faith to enhance their power and sway over the masses. This is true for Christians, Jews, Muslims, Hindus, … virtually every major faith. It is manifesting itself in the movement to authoritarian rule in the United States. The forces of hate are cloaked in faith to immunize themselves from critical views. Hatred, discrimination and bigotry are justified in their faith and freed from much critical feedback. They also complain about being repressed by society even when they are the majority and ruling social order. This is a common response when the forces of inclusion complain about their institutionalized bigotry. In the United States the minority that is truly oppressed are atheists. An atheist can’t generally get elected to office and in many places needs to hide this identity lest be subject to persecution. It is among the personal identities that I need to hide. At the same time Christians can be utterly open and brazen in self-expression of their faith.

In virtually every case the forces against inclusion are fear-driven. Many people are not comfortable with people who are different because of how it reflects on them. For example, many of the greatest homophobes actually harbor homosexual feelings of their own they are trying to squash. Openly accepted homosexuality is something that they resist because it seems to implicitly encourage them to act on their own feelings. In response to these fears they engage in bigotry. Generally, gender and sexual identities will bring these attitudes out because of the discomfort with possibilities the identities offer people. These sorts of dynamics are present with religious minorities too. Rather than question their faith, gender or sexuality, the different people are driven into their respective closets and out of view.

Getting at Subtle Exclusion

 ‘Controversial’ as we all know, is often a euphemism for ‘interesting and intelligent’.

― Kevin Smith

I’m writing this essay on diversity as a white middle-aged male who is seemingly a member of the ruling class, what gives? I represent a different diversity in many respects than the obvious forms we focus on. Being discriminated against happen without consequence. I’m outgoing and extroverted in a career and work environment dominated by introverts who are uncomfortable with emotion and human contact. I’m opinionated, outspoken and brave enough to be controversial in a sea of committed conformists. Both of these traits are punished with advice to blunt my natural tendencies. You are expected to toe the line and conform. The powers that be are not welcoming to any challenge or debate, the mantra of the day is “sit down, shut up, and do as you’re told” and “you’re lucky to be here”. The message is clear, you aren’t here as an equal, you need to be a compliant cog who plays your role. Individual contributions, opinions and talent are not important or welcome unless they fit neatly into the master’s plan. Increasingly in corporate America, you are either part of the ruling class or simply a disempowered serf whose personal value is completely dismissed. In this sense any diversity is discouraged and squashed by our overlords.

If one trait defines the vast number of people today, it is disempowerment. Your personal value and potential are meaningless to our management. Do you job; do what you are told to do; comply with their requirements. Whatever you do don’t make waves or question the powers. I’ve done this, and the reaction is to punish people through marginalization and lack of access. Retribution is implicit and subtle. In the end you don’t know how much you were punished through opportunity and information denial. The key is that the powers that be are in control of you and if you don’t play ball with them, you are an outsider. A way of not getting into trouble is being a compliant, boring and utterly vanilla worker bee. If you tow the party line you simply get the benefit of employment “success”. Eventually if you play your cards right you can join the power structure as one of them. In this way diversity of any sort is punished. They see diversity as a threat and work to drive it out.

You could jump so much higher when you had somewhere safe to fall.

― Liane Moriarty

Given that the treatment of relatively benign diversity is met with implicit resistance, one might reasonably ask how edgier forms of diversity will be welcomed? Granted, one’s personality can have a distinct work-related impact, and behavior is certainly work appropriate, but what are the bounds on how that is managed? My experience that the workplace goes too far in moderating people’s innate tendencies. My particular work has promoted the aspirational view toward diversity of bringing your true and best self to work. In all honesty, it is an aspiration that they fail at to a spectacular degree. When relatively common and universal aspects of our humanity are not accepted, how would more controversial aspects be accepted? A fair reaction is the conclusion that they wouldn’t be accepted at all. To add to the mix is a broader governance that is getting less accepting of differences than more. As a result, people whose lives are outside the conservatively defined norms are prone to hide major aspects of their identity. The accepted majority and accepted identity is white, male, Christian, heterosexual, and monogamous. We still accept political differences, but to varying degrees we expect people to fall into only a few narrow bins.

It is time for parents to teach young people early on that in diversity there is beauty and there is strength.

― Maya Angelou

What happens when someone falls outside these identities? If one is female and/or non-white there are protections and discrimination and bias is purely implicit. What about being an atheist? What about being gay, or transsexual? The legal protections begin to be weaker, and major societal elements seek to remove these protections. What if you’re non-monogamous, or committed to some other divergent sexual identity? What if you’re a communist or fascist? Then you are likely to be actively persecuted and discriminated against. Where is the line where this should happen? What life choices are so completely outside the norm of societal acceptance that they should be subject to effective banishment. If you have made one of these life choices, your choice is often hidden and secret. You are in the closet. This is a very real issue especially when large parts of society want you in the closet or put you back there. Moreover, this part of society is emboldened to try and put gays, women and people of color back in their traditional closets, kitchens and ghettos.

We are in a time where progress in diversity has stopped, and elements of society are moving backwards. Progress in expanding diversity and acceptance has stopped, and we are making America Great Again by enhancing the power of whites hiding people of color, closeting gays, and making women subservient. All of this is done in service of those in power both to maintain their power, wealth and control. Any sort of commitment to diversity is viewed as an assault on the power and wealth of the ruling class. As a result, we see a continued concentration of wealth and power in the hands of the few. General social mobility has been diminished so that people keep their place in society. For the rank and file, we see disempowerment and meager rewards for slavish conformity. Step out of line and draw attention to yourself and expect punishment and shunning. The slavishly conformant masses provide the peer pressure and bulk of the enforcement, but all of it serves those in power.

Instead of creating a society and system that gets the best out of people and maximizes human potential, we have pure maintenance of power by the ruling class. We let people act out through their basest fears to squash people and ideas that are uncomfortable. We are not leading people to be better, we are encouraging them to be worse. Rather than act out of love, we are emboldening hate. Rather than accept people and allow them flourish using their distinct individuality and achieve satisfying lives, we choose conformity and order. The conformity and order that is imposed serves only those in power and limits any aspirational dreams of the masses. The masses are controlled by fear that naturally arises with those who are different. Humans are naturally fearful of other humans that are different. Rather than encourage people to accept these differences (i.e., accept and promote diversity), we encourage them to discriminate against them by promoting fear of the other. This fear is the tool of rich and powerful to create systems that maintain their grip on society.

We are in need of leadership that blazes a trail to being courageous and better. Without leadership and bravery, the progress we have achieved will turn back and be lost. There is so much more to do to get the best out of people, which cannot happen when our leaders allow and even encourage the worst in people to serve their own selfish needs. We have a choice and it is past time to choose excellence and progress through inclusion and diversity.

 Here’s to the crazy ones. The misfits. The rebels. The troublemakers. The round pegs in the square holes. The ones who see things differently. They’re not fond of rules. And they have no respect for the status quo. You can quote them, disagree with them, glorify or vilify them. About the only thing you can’t do is ignore them. Because they change things. They push the human race forward. And while some may see them as the crazy ones, we see genius. Because the people who are crazy enough to think they can change the world, are the ones who do.

― Rob Siltanen

Your assumptions are your windows on the world. Scrub them off every once in a while, or the light won’t come in.

― Isaac Asimov

In conducting science, the importance of models is central to practice. Modeling is paired with observation as Man’s abstraction for understanding the World around us. Models need to be descriptive and tractable for examining nature.  These two aspects can be in direct conflict with each other. Observation under natural or controlled circumstances provides the core of scientific knowledge. Observation becomes science when we provide a systematic explanation for what we see. More often than not this explanation has a mathematical character as the mechanism we use. Among our mathematical devices differential equations are among our most powerful tools. In the most basic form these equations are rate of change laws for some observable in the World. Most crudely these rate equations can be empirical vehicles for taking observations into a form useful for prediction, design and optimization. A more basic form is partial differential equations (PDEs) that describe the basic physics in a more expansive form. It is important to consider the consequences of the model forms we use. Several important categories of models are intrinsically unphysical in aspects thus highlighting the George Box aphorism that “essentially, all models are wrong”!

Assumptions are the most damaging enemies of our mind’s equilibrium…An assumption is an imaginary truth.

― A.A. Alebraheem

Partial differential equations come in three basic flavors, hyperbolic, parabolic and elliptic. These flavors describe some of the basic character of the equations and have fundamental differences in how they are solved, understood as objects and more importantly physical context. The core of this essay is going to be physical in nature and to the point only hyperbolic equations are primal in physics. This is to say that at the basic level everything that we might describe as a physics law is hyperbolic in character. This is for a simple and very good reason, the principle of causality. Cause and effect, the flow of time and the presence of a cosmic speed limit. If we adhere to these maxims, the conclusion is utterly obvious. Other forms of PDEs produce instantaneous global effects that violate this principle. This in no way implies that parabolic or elliptic models are not incredibly useful, they are. Their utility and other properties exceed the issues with causality violations.

More on that point soon, but first a bit of digression on the other forms of PDEs. The classical elliptic equations is Laplace’s equation, . Elliptic equations are the simplest form and often describe physics where spatial terms are in equilibrium and there is no temporal, rate terms. Elliptic equations can include time terms, but usually implying something so deeply unphysical as to be utterly outlawed. If time is elliptic, the past is determined by the future, and since we know that time flows in one direction, this is deeply and fundamentally unphysical. In other uses, the elliptic PDEs are found through ignoring temporal terms. This is a philosophical violation of the second law of thermodynamics, which can be used to establish the arrow of time. In this sense we find that elliptic equations are an asymptotic simplification of more fundamental laws. Another implication of ellipticity of PDEs is infinite speed of information, or more correctly an absence of time. If elliptic equations are found within a set of equations, we can be absolutely sure that some physics has been chosen to be ignored. In many cases these ignored physics are not important and some benefit is achieved through the simplification. On the other hand, we shouldn’t lose sight of what has been done and its potential for mischief. At some point this mischief will become relevant and disqualifying.

Assumptions aren’t facts; they’re opportunities for research and testing.

― Laurie Buchanan

Next along the way we have parabolic equations and we can repeat the above discussion. Most classically the equation of heat transfer is parabolic (along with other diffusion processes). The classical form is the heat equations, . We often learn that these diffusion processes are fundamental, leading to the second law of thermodynamics. This comes with a deep problem that we should acknowledge. The parabolic equations imply an infinite propagation speed. Physically the process of diffusion is quite discrete associated with collisionality of the particles that make up materials, or discrete effects of solids (where electrons are particles that move, exchange and interact). This physical effect is utterly bound by finite speeds of propagation.

With elliptic equations the strength of the signal is unabated in time, but with parabolic equations, the signal diminishes in time. As such the sin of causality violation isn’t quite so profound, but it is a sin nonetheless. As before we get parabolic equations by ignoring physics. Usually this is a valid thing to do based on the time and length scales of interest. We need to remember that at some point this ignorance will damage the ability to model. We are making simplifications that are not always justified. This point is lost quite often. People are allowed to think the elliptic or parabolic equations are fundamental when they are not.

We now get to the third category of PDEs, the hyperbolic kind. The simplest form is a wave equation, . This can be written as a system of equations first-order PDEs, and . We can derive the simple wave equation by differentiating the first equation in time and the second in space then substituting to eliminate . The propriety of these steps depends on the variable being continuously differentiable, i.e., smooth. The second, first-order form is the entry point for the beautiful mathematics of hyperbolic conservation laws. As we will show, the elliptic and parabolic equations are simplifications of the hyperbolic equations made upon applying some assumptions.

I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail.

― Abraham H. Maslow

One key example of an equation where the presumption of the basic laws of physics is generally wrong are diffusion processes. One might consider Fourier’s law to be a fundamental law of physics as applied to heat condition in a parabolic form, . Instead this is a simplification of a more broadly valid law where heat flows using a hyperbolic equation. This requires a simple modification of the Fourier law to .  For most applications heat flow can be modeled in the parabolic form as the hyperbolic form is important over very short time and space scales. Still the more fundamental law is the hyperbolic form, and the classical parabolic form is derived by assuming that certain aspects of the dynamics can be ignored. We must always remember that the standard modeling of diffusion processes has an unphysical aspect baked into the equations.

The importance of the hyperbolic character of heat conduction may be more important. It is related to the property of material called second sound. This property is measured and known to be significant under cryogenic conditions where quantum effects are significant. It is also very hard to measure. The leading and rather compelling fact is its relation to the sound speed; the second sound is slower than the sound speed. This would mean that its effective time scale is also longer than the acoustic time. If this aspect is generally true, then the time scale can’t be ignored under many conditions. The deeper question is how all of this plays with thermodynamics. So much of the grounding of thermodynamics is in an equilibrium setting, and this phenomenon adds a natural and potentially important relaxation time scale.

There is an issue with hyperbolic diffusion that we should acknowledge. This form of the equations can violate the second law of thermodynamics, which underpins the macroscopic dynamics of the universe and the arrow of time. By the same token the imposition of the second law through a physical process in continuum physics is invariably tied to diffusion. As such we have formed a veritable technical Mobius strip.  A question is whether a fundamentally different equation can actually violate a less physical law that it is based upon. This might call the violations of the second law by hyperbolic diffusion into question rather directly! In other words, what would change about the second law of thermodynamics; if the diffusion process itself were hyperbolic. Perhaps this is a specific inroad to discussions of non-equilibrium thermodynamics. This may provide a necessary and distinct framing of a deeper discussion. Clearly infinite speeds of propagation for information is unphysical, functionally the second law could be recovered to account for temporal effects.

There is nothing so expensive, really, as a big, well-developed, full-bodied preconception.

― E.B. White

The incompressible Navier-Stokes equations are second primal example where hyperbolic equations are replaced by both parabolic-elliptic equations. A starting point would be the compressible equations that are purely hyperbolic without viscosity. Of course, the viscosity could be replaced with hyperbolic equations to make the compressible flow totally hyperbolic. These equations are the following, . Previously, we discussed the replacement of hyperbolic diffusion by the parabolic terms. For the incompressibility we remove sound waves analytically. The key to doing this is remove any connection between pressure and density with the divergence free constraint, . This also turns mass into a passively advected scalar. This is a useful model for low speed flows, but the diffusion and suppressed sound waves both produce infinite speeds of propagation. This violates the principle of causality where there is cause and effect. Instead everywhere is impacted by everything immediately.

As noted repeatedly these infinite speeds are definitely and unabashedly unphysical and signs that the equations are intrinsically limited in modeling scope. These issues are almost routinely ignored by most scientists and engineers. The reason is that the assumptions associated with parabolic or elliptic equations are valid for use. This will not always be true. It should be in the back of their mind. The message is clear, the equations will become invalid under some conditions, some length or time scale will unveil this invalidity. The question is what are these scales, and have we stumbled upon them yet? More generally the use of parabolic or elliptic equations produce these unphysical effects as a matter of course. This implies that the model equation will lose utility at some point under some conditions. We simply need to guard ourselves to this potential and keep this firmly in the back of our mind. The issue in this regard is the lack of capability to solve these non-standard models and make a complete assessment of model validity. By the same token, the non-standard models are harder to solve and may have deleterious side-effects if the full physics is retained.

A very good example of these side effects occurs with compressible flows when the Mach number is small. Solving low-Mach number flows with compressible codes is terribly inefficient and prone to significant approximation errors. This has a great deal to do with the separation of scales. As a result, the solutions often do not adhere to expectations. The consequence is there are many “fixes” to compressible flow solvers to remove the difficulties. The odd thing about this issue is the definitely greater physical reality associated with the compressible flow equations as compared to the incompressible equations. This might imply that the conditioning of the equations is the greatest problem. In addition, modern shock capturing methods have an implied discontinuity with their construction. It would seem that a continuous approximation might alleviate the problems. Conditioning issues with the separation of scales remains.

For modeling and numerical work, the selection of the less physical parabolic and elliptic equations provides better conditioning. The conditioning provides a better numerical and analytical basis for the solutions. The recognition that the equations are less physical is not commonly appreciated. A broader and common appreciation may provide impetus for identifying when these differences are significant. Holding models that are unphysical as sacrosanct is always dangerous. It is important to recognize the limitations of models and allow ourselves to question them regularly. Even models that are fully hyperbolic are wrong themselves, this is the very nature of models. By using hyperbolic models, we remove an obviously unphysical aspect of a given model. Models are abstractions of reality, not the operating system of the universe. We must never lose sight of this.

Everything must be made as simple as possible. But not simpler.

― Albert Einstein

Courant, Richard, and David Hilbert. Methods of mathematical physics [Methoden der mathematischen Physik, engl.] 1. CUP Archive, 1965.

Lax, Peter D. Hyperbolic partial differential equations. Vol. 14. American Mathematical Soc., 2006.

Körner, C., and H. W. Bergmann. “The physical defects of the hyperbolic heat conduction equation.” Applied Physics A 67, no. 4 (1998): 397-401.

Chester, Marvin. “Second sound in solids.” Physical Review131, no. 5 (1963): 2013.

Christov, C. I., and P. M. Jordan. “Heat conduction paradox involving second-sound propagation in moving media.” Physical review letters 94, no. 15 (2005): 154301.

Fefferman, Charles L. “Existence and smoothness of the Navier-Stokes equation.” The millennium prize problems 57 (2006): 67.

Doering, Charles R. “The 3D Navier-Stokes problem.” Annual Review of Fluid Mechanics 41 (2009): 109-128.

… Nature almost surely operates by combining chance with necessity, randomness with determinism…

― Eric Chaisson

On many occasions I’ve noted the tendency for science to see the World through a highly deterministic lens. We do this despite the World around us that includes a large degree of chance and random events. In science we might consider highly deterministic experiments to be well designed and useful. In a sense this is correct as such experiments confirm our existing theories grounded heavily in determinism. When we take this attitude into the real World of observation of nature, or engineered systems, the deterministic attitude runs aground. The natural World and engineered systems rarely behave in a completely deterministic manner. We see varying degrees of non-determinism and chance in how things work. Some of this is the action of humans in a system, some of it are complex initial conditions, or structure that deterministic models ignore. This variability, chance, and structure is typically not captured by our modeling, and as such modeling is limited in utility for understanding reality.

The assumption of an absolute determinism is the essential foundation of every scientific enquiry.

― Max Planck

Determinism. universe has a starting point (Big Bang?) correct formulations for laws of nature allow histories of all particles to be traced and predicted into the future. everything is predictable, universe functions like clockwork. Free will? Sir Isaac Newton.

Mathematical models of reality are heavily grounded in a deterministic assumption. This grounding is largely the legacy of Newton whose assumptions were heavily influenced by his religious faith and an almighty God. This God controlled the universe and determined the outcomes. These beliefs ran headlong into reality in the 20^th Century with quantum physics and the need for probabilities in models. The power of non-determinism for the most fundamental laws of physics was undeniable, but at larger scales determinism rules supreme. We explain that the law of large numbers pushes the laws of physics over into determinism. On the other hand, we have pervasive laws like the second law of thermodynamics that encapsulate the disorder in the World in the deterministic view. Is this sufficient to capture all of non-determinisms role? I think not. In this sense the work of Newton and 19^th Century thought still controls much of science today. Almost every modeling exercise is following determinism as an unspoken underlying assumption. This happens without regard to what we see each day in the real World. The second law of thermodynamics and the power of entropy is not adequate to capture the full span of disorder’s impact on our World. This assumption does untold damage, and it is time to overthrow some aspects of determinism as it has outlived its utility.

We ought to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow. An intelligence knowing all the forces acting in nature at a given instant, as well as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were sufficiently powerful to subject all data to analysis; to it nothing would be uncertain, the future as well as the past would be present to its eyes. The perfection that the human mind has been able to give to astronomy affords but a feeble outline of such an intelligence.

― Pierre-Simon Laplace

Complex systems and experiments have a great deal of non-determinism in their fundamental behavior and outcomes.   Commonly this non-determinism is completely ignored and modeled with a fully deterministic modeling approach (e.g., the second law). More to the point, a better assumption is a combination of deterministic and stochastic effects are present. The stochastic effects are largely ignored today and swept up into the deterministic model in a heavy handed one-size fits all manner. This sort of approach isn’t usually even considered as a problem because the behavior is considered totally deterministic. The consequence is an inability to consider the proper source of the behavior is misattribution. We are putting physical effects that are non-deterministic into a model that is deterministic. This should seriously limit the predictive power of our modeling.

To move forward we should embrace some degree of randomness in the fundamental models we solve. This random response naturally arises from various sources. In our deterministic models, the random response is heavily incorporated in boundary and initial conditions. The initial conditions include things like texture and structure that the standard models homogenize over. Boundary conditions are the means for the model to communicate with the broader world, which has vast complexities are grossly simplified. In reality both the initial and boundary conditions are far more complex than our models currently use.

The sort of deterministic models we use today attempt to include the entire system without explicitly modeling non-deterministic aspects. These effects are incorporated into the deterministic model or end up increasing the uncertainty of the modeling effort. Our efforts could advance significantly by directly modeling the stochastic aspects. This would produce an ability to separate the modeling effects that are completely deterministic from those that are random plus the interaction between these. We might expect that producing models with an appropriate separation would make the deterministic modeling lower in uncertainty. Some amount of uncertainty in any of these systems is irreducible, and proper modeling of the non-deterministic would produce results that capture these effects properly. Instead of being irreducible this aspect would simply be part of the model, and part of the result. It would move from being uncertain to being part of the answer. We should not expect that modeling non-deterministic dynamics with deterministic models to be the best we can do.

Applying logic to potentially illogical behavior is to construct a house on shifting foundations. The structure will inevitably collapse.

― Stewart Stafford

Another aspect of the complexity that current modeling ignores are the dynamics associated with the stochastic phenomena or lumps it whole cloth into the model’s closure. In a real system the stochastic aspects of the model evolve over time including nonlinear interactions between deterministic and stochastic aspects. When the dynamics are completely confined to deterministic models, these nonlinearities are ignored or lumped into the deterministic mean field. When models lack the proper connection to the correct dynamics, the modeling capability is diminished. The result is greater uncertainty and less explanation of what is happening in nature. From an engineering point of view, the problem is that the ability to explicitly control for the non-deterministic aspect of systems is diminished because its influence on results isn’t directly exposed. If the actual dynamics were exposed, we could work proactively to design better. This is the power of understanding in science; if we understand we can attempt to mitigate and control the phenomena. Without proper modeling we are effectively flying blind.

Die Quantenmechanik ist sehr achtung-gebietend. Aber eine innere Stimme sagt mir, daß das doch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, daß der nicht würfelt.

Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the “old one.” I, at any rate, am convinced that He does not throw dice.

–Albert Einstein, in a letter to Max Born

Today’s current modeling paradigm is relentlessly deterministic. We attempt to model experiments as a single well-determined event even when significant aspects of the experiment are non-deterministic. Effectively the non-deterministic aspects are ignored or misattributed to determinism. The second experiment is then inconsistent because it is a different instance, instead of being the same deterministic case with a different stochastic forcing. If we model the stochastic element of the phenomena directly we can get to understanding its impact. With our current modeling we simply drive a fundamental misunderstanding of what is happening. We are left with models that have fundamental limitations. None of these issues is going to be handled by brute force. Neither computer power, or computational accuracy, or algorithmic efficiency will impact these problems. The answer to the issues are centered on modeling and increasing its span of physical phenomena addressed.

The impediments to changing our modeling are massive. We have a strong tendency to lump all of the non-deterministic effects into constitutive laws and closure that lends itself to relative ease of modification. Changing or expanding the governing equations in a code can be utterly daunting and usually not supported by current funding. The entire enterprise of developing new equations is difficult and risky in nature. Our system today is utterly opposed to anything risky and actively undermines attempting anything difficult at every turn. Our computational science is extremely invested in existing models, and most paths to improvement are routed through them. Increasingly we are invested in the most brutish and painfully naïve path to improvement by investing almost entirely in faster computers. The most painful aspect of this path is its lack of timeliness, the ease of creating faster computers has ended with the death of Moore’s law. Getting a faster computer is now extremely expensive and inefficient. Other paths to improvement are not favored and we have almost forgotten how to do science like that. The capper to this sad tale is the utter inability of these computers to help fix faulty models. We have lost the ability to conduct intellectually rigorous work.

The sort of science needed is enormously risky. I am proposing that we have reached the end of utility for models used for hundreds of years. This is a rather bold assertion on the face of it. On the other hand, the models we are using have a legacy going back to when only analytical solutions to models were used, or only very crude numerical tools. Now our modeling is dominated by numerical solutions, and computing from the desktop (or handheld) to supercomputers of unyielding size and complexity. Why should we expect models derived in the 18^th and 19^th centuries to still be used today? Shouldn’t our modeling advance as much as our solution methods have. Shouldn’t all the aspects of modeling and simulation be advancing. The answer is a dismal no.

The reasons for this dismal state of affairs is somewhat reasonable. The models defined over the past few centuries defined general solution. Computing offered a path to solution that analytical methods failed to provide. As a result, we saw computing work to provide useful solutions to models that have had limited utility for a very long time. The models now being routinely solved numerically that had been unavailable for a huge amount of time. The numerical work is often done quite poorly with marginal quality control. Assessment of the quality of numerical work is usually slipshod and casual. The “eyeball” and “view graph” norm rule quantified uncertainty and error. Most good results using these models are heavily calibrated and lack any true predictive power. In the absence of experiments, we are generally lost and rarely hit the mark. Instead of seeing any of this as shortcomings in the models, we seek to continue using the same models and focus primarily on computing power as a remedy. This is both foolhardy and intellectually empty if not outright dishonest.

As such the evidence that our models are inadequate is overwhelming. Our response to this evidence has been virtual ignorance of this conclusion. We continue to invest in the same areas that have failed to improve results over a long period of time. We continue to sell massive computing power as the fix it all, remedy for problems. We fail to recognize that neither computing power, or solution accuracy will cure any problems if the fundamental model is flawed. Our fundamental models are flawed, and the routes taken for improving modeling and simulation will not help. If the basic model has flaws, a faster computer or a better method, a more accurate discretization, or better scaling will not help. The only cure is to fix or change the model. One of the biggest places where modeling fails is modeling the separation in deterministic and non-deterministic aspects of our models.

A simple setting and familiar setting to see how this might help is weather. If we look at any of our models, at any of our scales, it is obvious that enormous variability and details are being excluded from our modeling. One of the biggest needs of weather modeling are extreme weather events that dominate financial and political consequences of weather. Analogous issues exist in a myriad of other fields where modeling and simulation impact the science. A reasonable supposition is that interactions among the averaged over and ignored fine scale structure help produce extreme events when interacting with the large-scale weather. It is well known that large scale weather phenomena set the stage or increases the likelihood of extreme events. The actual phenomenology of extreme events depends on how the large-scale weather interacts with local detail such as the surface topology.

Analogous phenomena happen is many other fields such as material failure, and turbulence. These models are strained under the demands of the modern World and progress is desperately needed. The solutions are not being supported, instead focused on risk adverse and rather pedestrian approaches while eschewing riskier work like model creation. The focus on computing power reflects this intellectual cowardice quite acutely. Our current models are limited by their fundamental structure rather than solution methods, or computing power. Our science programs need to address these challenges in a credible manner by coupling a focus on theory with innovations in experimental science. The challenge is not refining old ideas but allowing ourselves to pursue new ones with sufficient freedom and aggression. Our greatest challenge is not the science, but rather our inability to conceive of solutions in today’s World. This work could be enormously valuable to society as a whole if we could envision it and take the risks necessary to reach success.

The difficulty lies not so much in developing new ideas as in escaping from old ones.

― John Maynard Keynes