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Love Is Not Enough: Mathematics Of Love

May 15, 2010

Life is beautiful, but it’s love that gives meaning to life. Everyone is craving for that perfect everlasting love. People find their love, commit to each other for life with the belief that it’s going to last for ever and their love is sufficient to cruise them into this perfect everlasting relationship for life. And why not, the relationship is based on Love so it should be enough, you think so, right. Then, why so many breakups, so many divorces (especially in US and Europe). Is Love Enough to sustain a relationship? Is their a recipe for a perfect stable relationship. You can get into philosophy of love and life and keep it discussing it for ever or you can formulate this unique emotion/sentiment based real life situation as a mathematical problem and try to find a solution by solving few equations. Well, José-Manuel Rey has done the latter to understand the dynamics of love and stability of relationship and that too in a very beautiful way. The paper is available free to download here.

Here is the Abstract of the paper:

Background: Marital dissolution is ubiquitous in western societies. It poses major scientific and sociological problems both in theoretical and therapeutic terms. Scholars and therapists agree on the existence of a sort of second law of thermodynamics for sentimental relationships. Effort is required to sustain them. Love is not enough.
Methodology/Principal Findings: Building on a simple version of the second law we use optimal control theory as a novel approach to model sentimental dynamics. Our analysis is consistent with sociological data. We show that, when both partners have similar emotional attributes, there is an optimal effort policy yielding a durable happy union. This policy is prey to structural destabilization resulting from a combination of two factors: there is an effort gap because the optimal policy always entails discomfort and there is a tendency to lower effort to non-sustaining levels due to the instability of the dynamics.
Conclusions/Significance: These mathematical facts implied by the model unveil an underlying mechanism that may explain couple disruption in real scenarios. Within this framework the apparent paradox that a union consistently planned to last forever will probably break up is explained as a mechanistic consequence of the second law.

There are two important variables in this model. First one is the ‘feeling variable’ x (t) which represents the state of relationship or the common sentiments towards each other. When people fall in love, they have a very strong feeling towards each other. At the beginning of relationship, this common feeling towards each other is assumed in this model to be very large, x(0). According to Second law of thermodynamics for sentimental relationships, there is tendency for the initial feeling for one another to fade away. This kind of inertia must be counteracted by conscious practices. So applying this second law, it can be assumed that x(t) will decrease in a relationship with the passage of time and x(0) > x(t) at all the times. There is a critical value of x, x(min), at which relationship will fall apart.

Coming to second variable, its the effort variable, c(t), which is the amount of effort required in day to day activities, which can be a small gesture or a big sacrifice, in order to counter the fading sentiment. This effort variable, reinforces the relationship on day to day basis. But this effort comes at a cost to the individual and there is a maximum amount of effort a person can apply before it starts taking toll on the person and it becomes too much for the person to handle the energy associated with the efforts. In terms of the equation this second law of relationship can be written as

In this state equation, c(t) effort variable, is the control variable, while x(t) the feeling variable is the controlled variable. Here a is the effort efficiency factor, meaning how effective is your effort. If there is no effort in replenishing your relationship, then relationship continually degrades and ultimately breakups.

Speaking in simple words, when you make efforts, you make sacrifices to make your partner happy, and that makes you happy in return. But if the efforts required are too high for the emotional comfort level of the individual, these extra efforts can start bringing stress in the person as well as in the relationship. So there is a maximum effort level ,c(*), beyond which extra effort doesn’t help in rebuilding the relationship, also can be called as maximum tolerable effort level.

What you see in this plot is that a sentimental equilibrium solution is possible shown as point E, given that the feeling variable x>x(min) which ofcourse is required and easy to understand. The second requirement for this equilibrium is that the effort level c> c(*), i.e. c-c(*) >0, meaning you need more efforts than what is perceived as comfortable or tolerable effort level by the individuals. In other words, individuals in relationship have to be ready to make that extra effort to go beyond their tolerable effort level and fill that “effort gap” to make relationship stable. Failure to do so can lead to demise of the relationship.

So summarizing, when in relationship you start with a mutual emotional feeling of well being for each other at say x(0). With time it starts fading away, but you make efforts to replenish the relationship by making that extra effort and keep your relationship status at pint A in the above plot. Ideally , you have to reach to point E which is the equilibrium, so you need continuously increasing efforts. But suppose, you relaxed for a while in making those efforts, so you fall down a bit due to these ‘effort inattentions’ as shown in the plot. But as sincere couples you realize and get back to work it out again quickly, so you are back in track to point E, but now you have to put in more efforts. If you keep being inattentive to these efforts, you can completely lose your track and the feeling will keep on fading and you can ultimately hit x(min), the point where the feeling is totally gone, and ultimately can lead to breakups or divorce.

Sadly, this study, while beautiful mathematically, doesn’t give a rosy picture of stability of love and feelings in a relationship. Love is not sufficient, you need to nurture it, every moment, everyday and still there can be factors and perturbations in life, which can throw your relationship offtrack. But its not bleak altogether, atleast there is hope, if you can find someone who is willing to work together making these efforts continuously, Life can Be Beautiful.

PS: Will this model be applicable for relationships in India, where not just the two individuals play a role in relationship, rather the whole family and friend circle is involved which can be a stabilizing factor sometimes, but can also be a destabilizing factor.

Reference: Rey J-M (2010) A Mathematical Model of Sentimental Dynamics Accounting for Marital Dissolution. PLoS ONE 5(3): e9881. doi:10.1371/ journal.pone.0009881

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